diff --git "a/ANAzT4oBgHgl3EQfvv6Q/content/tmp_files/load_file.txt" "b/ANAzT4oBgHgl3EQfvv6Q/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/ANAzT4oBgHgl3EQfvv6Q/content/tmp_files/load_file.txt" @@ -0,0 +1,1275 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf,len=1274 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='01712v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='PR] 4 Jan 2023 1 Mesoscopic eigenvalue statistics for Wigner-type matrices Volodymyr Riabov∗ Institute of Science and Technology Austria volodymyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='riabov@ist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='at Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We prove a universal mesoscopic central limit theorem for linear eigenvalue statistics of a Wigner- type matrix inside the bulk of the spectrum with compactly supported twice continuously differentiable test functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The main novel ingredient is an optimal local law for the two-point function T(z, ζ) and a general class of related quantities involving two resolvents at nearby spectral parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Date: January 5, 2023 Keywords and phrases: Wigner-type matrix, mesoscopic eigenvalue statistics, central limit theorem 2010 Mathematics Subject Classification: 60B20, 15B52 1 Introduction In the study of the eigenvalue distribution of large random matrices, the most celebrated analog of the Law of Large Numbers is the Wigner semicircle law [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' It states that the empirical density of eigenvalues converges to a deterministic limit known as the semicircle distribution ρsc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' More explicitly, if H is an N ×N Wigner matrix and f is a sufficiently smooth test function, then the linear eigenvalue statistics N −1 Tr f(H) converge in probability to � R f(x)ρsc(x)dx in the large N limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The corresponding Central Limit Theorem (CLT) asserts that the asymptotic fluctuations of the linear eigenvalue statistics Tr f(H)−E [Tr f(H)] are Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The absence of the N −1/2 normalization factor, appearing in the classical CLT, can be viewed as a manifestation of the strongly-correlated nature of the eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' For the special case of f(x) = (x − z)−1 with Im z ̸= 0, this result was obtained by Khorunzhy, Khoruzhenko and Pastur [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Johansson obtained the CLT for invariant ensembles with arbitrary polynomial potentials in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In [4], Bai and Yao used martingale CLT to establish the result for Wigner matrices with analytic test functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The proof for bounded test functions f with bounded derivatives appeared in the work of Lytova and Pastur [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In subsequent works, different moment conditions on the matrix and regularity conditions on the test function were studied extensively by many authors, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', [6, 18, 23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' While fixed test functions represent macroscopic averaging in the spectrum, one can introduce N- dependent scaling and consider scaled test functions of the form f(x) = g(η−1 0 (x − E0)), where E0 is a fixed reference energy in the bulk, η0 ≡ η0(N) ≪ 1 is a scaling parameter, and g is compactly supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Then Tr f(H) involves only about Nη0 eigenvalues of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In particular, on mesoscopic scales, corresponding to N −1 ≪ η0 ≪ 1, the limiting variance is given by the square of the ˙H1/2 norm of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Mesoscopic test functions were first studied by Boutet de Monvel and Khorunzhy in [7] for the Gaussian Orthogonal Ensemble, with subsequent extension to real Wigner matrices in [8] with N −1/8 ≪ η0 ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In [13], He and Knowles proved the CLT for Wigner matrices with general mesoscopic test functions for all scaling parameters N −1 ≪ η0 ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' ∗Supported by the ERC Advanced Grant ”RMTBeyond” No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 101020331 The result was extended to ensembles of greater generality in the more recent works, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', [5] and [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In particular, Li and Xu obtained mesoscopic CLT for generalized Wigner matrices 1 in the bulk and at the spectral edge with C2 c test functions in the full range of scales [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Finally, Landon, Lopatto, and Sosoe proved the bulk CLT for the much more general ensemble of Wigner-type matrices in [17] for two classes of C∞ test functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' For a special class of globally supported regularized bump functions , the proof is performed via resolvent techniques for large scales and extended to the entire mesoscopic range using Dyson Brownian motion (DBM) dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' For the more conventional compactly supported scaled test functions, the bulk CLT is established on all mesoscopic scales N −1 ≪ η0 ≪ 1 using a combination of DBM and Green’s function comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Wigner-type matrices were first introduced in [2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' they have centered entries Hjk independent up to the symmetry constraint H = H∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The matrix of variances S, defined by Sjk := E � |Hjk|2� , is assumed to be flat, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', Sjk ∼ N −1 and satisfy a piece-wise H¨older regularity condition (see (B)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' As the main step towards CLT in the present paper, we prove the optimal averaged and entry-wise local laws (Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3) for the two-point function T , defined by Txy(z, ζ) := � a̸=y SxaGay(z)Gya(ζ), x, y ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', N}, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1) where G(z) is the resolvent of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The corresponding result in the simpler setting of generalized Wigner matrices was obtained in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Using the optimal local law for T (z, ζ), we prove the bulk mesoscopic CLT for Wigner-type matrices in the full range of scales N −1 ≪ η0 ≪ 1 for compactly supported C2 scaled test functions (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Our proof relies entirely on resolvent methods, circumventing the DBM dynamics used in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Understanding T (z, ζ) is the crucial ingredient for the CLT as it was realized in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In fact, a suboptimal entry-wise local law for Txy(z, ζ) was proved in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 of [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' If one relies solely on resolvent methods, this local law provides sufficient control for mesoscopic CLT only on scales η0 ≫ N −1/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The main reason for this limitation is that the error term in [17] contains the norm of the inverted stability operator (defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In the present paper, we show that this factor can be removed by separating the destabilizing eigendirection corresponding to the smallest eigenvalue of the stability operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Using this method, we prove a local law for a general class of quantities involving two resolvents (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2) and deduce the optimal averaged and entry-wise local laws for T (z, ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In particular, this allows us to obtain the CLT on all mesoscopic scales without relying on DBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The main difficulty lies in the fact that the deterministic approximation of the resolvent for Wigner- type matrices is not a multiple of the identity matrix, contrary to the generalized Wigner case [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Consequently, the destabilizing direction is no longer parallel to the vector of ones, and generally, no closed-form expression is known for the corresponding eigenprojector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' It is important to note that for the deformed Wigner matrices studied in [20], the deterministic approximation is also not a multiple of the identity, but Sjk = N −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Therefore, the two-point function can be expressed as the square of the resolvent and can be studied using the local law, similarly to the standard Wigner case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Instead of approximating the destabilizing direction to circumvent this difficulty, we use a contour integral representation for the eigenprojector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' It allows us to extend the decomposition approach of [21] to the Wigner-type ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' This method benefits from yielding an integral representation for the variance on all mesoscopic scales, under weaker regularity conditions on the test function than in [17], and relying only on resolvent methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The paper is organized in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Section 2 contains the precise definition of the model and the statement of our main mesoscopic CLT result, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In Section 3, we present our main technical result, the optimal local law for two-point functions in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In Section 4, we collect notations and preliminary results to which we refer throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In Section 5, we deduce Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 from Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2, and prove Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 using a local law for T (z, ζ) (Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3) as an input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The proofs of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3 are presented in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In Section 7, we prove Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2, which relates the variance of the linear eigenvalue statistics to the ˙H1/2-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' I would like to express my gratitude to L´aszl´o Erd˝os for suggesting the project and supervising my work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' I am also thankful to Yuanyuan Xu and Oleksii Kolupaiev for many helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 1Generalized Wigner matrices are characterized by a flat doubly-stochastic matrix of variances S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Unlike the Wigner case, the entries Sjk are not assumed to be equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The limiting eigenvalue distribution remains semicircular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 2 2 Model and Main Result We begin with the definition of Wigner-type matrices originally introduced in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 of [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 (Wigner-type matrices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Let H = (Hjk)N j,k=1 be an N × N matrix with independent entries up to the Hermitian symmetry condition H = H∗ satisfying E [Hjk] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1) We consider both real and complex Wigner-type matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In case the matrix H is complex we assume additionally that Re Hjk and Im Hjk are independent and E[H2 jk] = 0 for k ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Denote by S the matrix of variances Sjk := E[|Hjk|2], and assume it satisfies cinf N ≤ Sjk ≤ Csup N , (A) for all j, k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', N} and some strictly positive constants Csup, cinf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We assume a uniform bound on all other moments of √ NHjk, that is, for any p ∈ N there exists a positive constant Cp such that E � | √ NHjk|p� ≤ Cp (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2) holds for all j, k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Additionally, we assume that S satisfies a H¨older regularity condition1, that is, |Sjk − Sj′k′| ≤ L N �|j − j′| + |k − k′| N �1/2 , (B) for all j, j′, k, k′ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', N} and some positive constant L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The constants cinf, Csup, Cp and L are independent of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 Central Limit Theorem for Mesoscopic Linear Eigenvalue Statistics Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5 in [17]) Let g be a C2 c (R) test function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Let ε0 be a small fixed constant and let N −1+ε0 ≤ η0 ≤ N −ε0, and let E0 be a fixed reference energy in the bulk of the spectrum, that is, ρ(E0) ≥ ε0 (here ρ is the density of states to be defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3) below ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Define the scaled test function f to be f(x) := g �x − E0 η0 � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3) then Tr f(H) − E [Tr f(H)] d−→ N � 0, 1 2βπ2 ∥g∥2 ˙H1/2 � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4) where β = 1 and β = 2 corresponds to real symmetric and complex Hermitian H, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We remark that the universal limiting variance in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4) coincides with the corre- sponding formulas for standard Wigner matrices [13], where Sjk = N −1, mj(z) = msc(z) for all j, k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' , N}, and msc(z) is the Stieltjes transform of the semicircle law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 1As stated in [2], assumption (B) can be weakened to piece-wise 1/2-H¨older regularity condition for some positive constant L on finitely many intervals, in the sense that max a,b max j,j′∈(NIb) max k,k′∈(NIa) N3/2 |Sjk − Sj′k′| |j − j′|1/2 + |k − k′|1/2 ≤ L, where {Ia}n a=1 is a fixed finite partition of [0, 1] into smaller intervals, and (NIa) denotes the set of positive integers j such that j/N lies in Ia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 3 3 Local Laws for the Two-point Functions In this section, we introduce our main technical result, local laws for quantities that involve two resolvents of a Wigner-type matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Our prime motivation is to study the function T (z, ζ) defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1), but our methods allow us to estimate a more general class of quantities, namely � a̸=y waGαa(z)Gaβ(ζ), � b � a̸=b WabGba(z)Gab(ζ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1) for fixed indices α, β, y, and deterministic weights wa, Wab satisfying |wa|, |Wab| ≤ cN −1 for some constant c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Here G(z) := (H − z)−1 denotes the resolvent of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Objects of this type were first studied in [11] in the setting of random band matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We obtain the estimates in the sense of stochastic domination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 in [12]) Let X = X (N)(u) and Y = Y(N)(u) be two families of random variables possibly depending on a parameter u ∈ U (N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We say that Y stochastically dominates X uniformly in u if for any ε > 0 and D > 0 there exists N0(ε, D) such that for any N ≥ N0(ε, D), sup u∈U(N) P � X (N)(u) > N εY(N)(u) � < N −D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We denote this relation by X ≺ Y or X = O≺(Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We consider spectral parameters z lying in the domain D, defined by D := {z ∈ C : N −1+τ ≤ | Im z| ≤ τ −1, | Re z| ≤ τ −1}, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2) for a fixed τ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' As in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2, our analysis is limited to the bulk of the spectrum, which we define via the self-consistent density of states ρ(E) ≡ ρN(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The density ρ(E) is recovered by the Stieltjes inversion formula, ρ(E) := π−1 lim η→+0 Im m(E + iη), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3) where m(z) := N −1 �N j=1 mj(z), and m(z) = (mj(z))N j=1 is the unique (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 in [2]) solution to the vector Dyson equation −1 m(z) = z + Sm(z), Im m(z) Im z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4) Let I be the set on which ρ(E) is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 of [2] guarantees that I consists of a finite union of open intervals (a(j), b(j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Then for κ > 0, we define the bulk domain by Dκ := {z ∈ D : Re z ∈ Iκ}, Iκ := � j [a(j) + κ, b(j) − κ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5) In particular, for all z ∈ Dκ, ρ(z) ≥ C(κ) for some constant C(κ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Given E0 as in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2, we choose κ so that E0 ∈ I2κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' There exists a positive constant ǫ = ǫκ which is independent of N, such that for all z, ζ in Dκ with | Re ζ − Re z| ≤ ǫ, and deterministic vectors w ∈ CN satisfying ∥w∥∞ ≤ cN −1, the following estimate holds, � a̸=y waGαa(z)Gaβ(ζ) = δαβ � m(z)m(ζ) � 1 − Sm(z)m(ζ) �−1w � α − δαβδαy[m(z)m(ζ)w]α + O≺ � (Ψ(z) + Ψ(ζ))(Ψ(z)Ψ(ζ) + 1{Im z Im ζ<0} min{Θ(z), Θ(ζ)}) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6) where the vector m is identified with the diagonal operator diag (m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Under the same conditions on z, ζ, for any deterministic N ×N matrix W satisfying |Wab| ≤ cN −1 for all a, b, the following estimate holds, � b � a̸=b WabGba(z)Gab(ζ) = Tr � m(z)m(ζ)Sm(z)m(ζ) � 1 − Sm(z)m(ζ) �−1W � + NO≺ � (Ψ(z) + Ψ(ζ))Ψ(z)Ψ(ζ) + 1{Im z Im ζ<0}Θ(z)Θ(ζ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7) 4 Here Ψ(z) and Θ(z) denote control parameters defined as Ψ(z) := � | Im m(z)| N|η| + 1 N|η|, Θ(z) := 1 N|η|, z = E + iη ∈ C\\R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='8) Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 implies the following averaged and entry-wise local laws for T (z, ζ) from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Let z, ζ satisfy the assumptions of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The entries Txy(z, ζ) admit the estimate Txy(z, ζ) = � (Sm(z)m(ζ))2 � 1 − Sm(z)m(ζ) �−1� xy + O≺ � (Ψ(z) + Ψ(ζ)) � Ψ(z)Ψ(ζ) + 1{Im z Im ζ<0} min{Θ(z), Θ(ζ)} �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9) Furthermore, for all deterministic N × N matrices A, the following equality holds Tr[A T (z, ζ)] = Tr[A � 1 − Sm(z)m(ζ) �−1� Sm(z)m(ζ) �2] + N ∥A∥ℓ∞→ℓ∞ O≺ � (Ψ(z) + Ψ(ζ))Ψ(z)Ψ(ζ) + 1{Im z Im ζ<0}Θ(z)Θ(ζ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='10) Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The error estimates in the entry-wise local law (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6), and hence in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9) are optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Indeed, for Sjk := N −1, which corresponds to the standard Wigner matrices, and ζ = ¯z, a simple calculation using the Ward identity shows that Txy(z, ¯z) = N −1| Im z|−1 Im msc(z) − N −1|msc(z)|2 + O≺ � Θ(z)Ψ(z) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='11) The error estimate in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7) is not optimal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' it can be improved to O≺ � N(Ψ(z) + Ψ(ζ))2� Ψ(z)Ψ(ζ) + 1{Im z Im ζ<0}NΘ(z)Θ(ζ) �� (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='12) However, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7) is sufficient for establishing the CLT, so for the sake of brevity, we do not present the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='12) in full detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We only indicate the necessary ingredients in Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='8 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 4 Notations and Preliminaries 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 Notations For a vector x = (xj)N j=1 ∈ CN we use the standard definitions of ℓ2 and ℓ∞ norms, namely, ∥x∥2 = � N � j=1 |xj|2 �1/2 , ∥x∥∞ = max j |xj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' For a linear operator T : CN → CN, we denote its matrix norms induced by ℓ2 and ℓ∞ norms, respectively, by ∥T ∥ℓ2→ℓ2 = sup ∥x∥2=1 ∥T x∥2 , ∥T ∥ℓ∞→ℓ∞ = sup ∥x∥∞=1 ∥T x∥∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' For two vectors x, y ∈ CN we use angle brackets to denote the ℓ2 scalar product, while for a single vector x ∈ CN angle brackets denote the average of its coordinates ⟨x, y⟩ = N � j=1 ¯xjyj, ⟨x⟩ = 1 N N � j=1 xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We use xy to denote a coordinate-wise product of vectors x and y, (xy)j = xjyj, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Similarly, for a given vector x with non-zero entries, 1x denotes a coordinate-wise multiplicative inverse � 1 x � j = 1 xj , j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 5 We use 1 to denote the vector of ones (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' , 1)t in CN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' For a measurable function f : R → R we use the standard definition of the Lp norms for p ≥ 1, and the following definition of the ˙H1/2 norm ∥f∥ ˙H1/2 = \uf8eb \uf8ed �� R2 |f(x) − f(y)|2 |x − y|2 dxdy \uf8f6 \uf8f8 1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' For two deterministic quantities X, Y ∈ R depending on N, we write X ≪ Y if there exists ε, N0 > 0 such that |X| ≤ N −ε|Y | for all N ≥ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Similarly, we write X ≲ Y if there exists a constant C, N0 > 0 such that |X| ≤ C|Y | for all N ≥ N0, and X ∼ Y if both X ≲ Y and Y ≲ X hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We use C and c to denote constants, the precise value of which is irrelevant and may change from line to line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 Local Law for the Resolvent In this subsection, we summarize the facts on Wigner-type matrices that we use throughout our proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Majority of these results were obtained in [1] (see also [3]), but we refer to their concise versions from [2] adapted for the Wigner-type setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 in [2]) The solution m(z) of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4) satisfies the following properties: (1) For every j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', N} there exists a generating probability measure νj(dx) such that mj(z) = � R νj(dx) x − z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1) (2) If the matrix of variances S satisfies conditions (A) and (B), then for all z ∈ C\\R, the solution admits the following bounds ∥m(z)∥∞ ≤ c 1 + |z|, ���� 1 m(z) ���� ∞ ≤ C(1 + |z|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2) We now state the optimal averaged and isotropic local laws for Wigner-type matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='8 in [2]) Let w, x, y be deterministic vectors in CN satisfying ∥w∥∞ = 1 and ∥x∥2 = ∥y∥2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Then the following estimates hold uniformly in z ∈ D: N −1��Tr � w(G(z) − m(z)) ��� ≺ Θ(z), ��⟨x, (G(z) − m(z))y⟩ �� ≺ Ψ(z), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3) where vectors m and w are associated with corresponding diagonal matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In particular, it follows from the isotropic local law (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3) that for any j, k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', N}, |Gjk(z) − δjkmj(z)| ≺ Ψ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3 Preliminary Bounds on the Stability Operator A significant part of our proof revolves around the stability operator, originally introduced in [1], that emerges when studying the two-point function T (z, ζ) defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In this subsection, we collect the known bounds on the stability and related operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The stability operator (1 − Sm(z)m(ζ)) is defined by the matrix with entries (1 − Sm(z)m(ζ))jk := δjk − Sjkmk(z)mk(ζ), j, k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', N}, z, ζ ∈ C\\R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5) Throughout this paper we use m (and various functions of m, such as Im m, |m|, m−1, m′) to denote both a vector (mj)N j=1 and the corresponding multiplication operator, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', diag � (mj)N j=1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Note that this notation agrees with the point-wise multiplication of two vectors if the first multiplicand is interpreted as an operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We stress which interpretation is used whenever ambiguity may arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 6 The analysis of the stability operator relies on the corresponding saturated self-energy operator F, studied in [17], that depends on two spectral parameters z, ζ, and is defined as Fjk(z, ζ) := |mj(z)mj(ζ)|1/2Sjk|mk(z)mk(ζ)|1/2, j, k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', N}, z, ζ ∈ C\\R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6) The following statements encompass the main properties of F and preliminary bounds on the stability operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3 in [17], c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9 and Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4 in [9]) For any z, ζ ∈ C, the principal eigenvalue of F defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6) is positive and simple, the corresponding ℓ2- normalized eigenvector v(z, ζ) has strictly positive entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The norm of F admits the following upper bound ∥F(z, ζ)∥ℓ2→ℓ2 ≤ 1 − 1 2 � | Im z| ⟨v(z, z), |m(z)|⟩ ⟨v(z, z), | Im m(z)| |m(z)| ⟩ + | Im ζ| ⟨v(ζ, ζ), |m(ζ)|⟩ ⟨v(ζ, ζ), | Im m(ζ)| |m(ζ)| ⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7) If |z|, |ζ| ≲ 1, then the entries of v(z, ζ) are comparable in size, that is cκ ≤ √ Nvj(z, ζ) ≤ Cκ, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', N}, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='8) and moreover, let Gap (F) denote the difference between the two largest eigenvalues of |F| = √ FF ∗, then Gap (F) admits the bound Gap (F) ≥ �δ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9) where �δ is a constant that depends only on the constants in conditions (A), (B) and κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Furthermore, for a fixed κ > 0 and z, ζ ∈ Dκ there exists a positive constant �cκ such that ∥F(z, ζ)∥ℓ2→ℓ2 ≤ 1 − �cκ (| Im z| + | Im ζ|) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='10) Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7 in [17]) Let z, ζ ∈ C, such that |z|, |ζ| ≲ 1 and Re z, Re ζ ∈ Iκ, then ��(1 − Sm(z)m(ζ))−1�� ℓ2→ℓ2 + ��(1 − Sm(z)m(ζ))−1�� ℓ∞→ℓ∞ ≲ 1 | Im z| + | Im ζ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='11) If additionally Im z Im ζ > 0, the estimate is improved to ��(1 − Sm �m)−1�� ℓ∞→ℓ∞ ≤ Cκ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='12) where Cκ > 0 is a positive constants dependent on κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Finally, we state the bounds on the stability operator in the special case of ζ = z, which is related to the derivative of m via the (vector) identity m′(z) = (1 − m2(z)S)−1m2(z), obtained by taking the derivative of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9 in [1], Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 in [9]) Let C > 0 be a positive constant, then for z ∈ C\\R with |z| ≤ C we have ��(1 − m2(z)S)−1�� ℓ2→ℓ2 + ��(1 − m2(z)S)−1�� ℓ∞→ℓ∞ ≲ |ρ(z)|−2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='13) where ρ(z) = π−1⟨Im m(z)⟩ is the harmonic extension of ρ(E) defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Therefore for all z ∈ C\\R with Re z ∈ Iκ we have ∥m′(z)∥∞ ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='14) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4 Cumulant Expansion Formula Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (Section II in [7], Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 in [13]) Let h be a real-valued random variable with finite moments, let f be a C∞(R) function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Then for any ℓ ∈ N the following expansion holds, E [h · f(h)] = ℓ � j=0 1 j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='c(j+1)(h) E � dj dhj f(h) � + Rℓ+1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='15) 7 where c(j) is the j-th cumulant of h defined by c(j)(h) = (−i)j dj dtj � log E � eith������ t=0 , and the remainder term Rℓ+1 satisfies |Rℓ+1| ≤ Cl E � |h|ℓ+2� sup |x|≤M |f (ℓ+1)(x)| + Cl E � |h|ℓ+2 · 1|h|>M � ���f (ℓ+1)(x) ��� ∞ , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='16) for any M > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We apply formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='15) with h equal to the matrix element Hjk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Correspondingly, in the real case (β = 1), C(p) denotes the matrix of p-th cumulants of H, C(p) jk := C(p)(Hjk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In the complex case (β = 2), C(p) is used as a notational shortcut and denotes the sum of matrices of p-th cumulants of real and imaginary parts of H, that is C(p) jk := C(p)(Re Hjk) + C(p)(Im Hjk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 5 Proof of the Main Result Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We divide the proof into two parts contained in the following propositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We indicate their analogs in the settings of [21] and [17] in parenthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 in [21] and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='76) in [17]) Let η0, ε0 > 0 and E0 satisfy the assumptions of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2, let f be a scaled test function defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3), and let φ(λ) be the characteristic function of Tr f(H) − E [Tr f(H)], φ(λ) := E [exp{iλ (Tr f(H) − E [Tr f(H)])}] , λ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1) Then its derivative φ′(λ) satisfies the following equation, φ′(λ) = −λφ(λ)V (f) + O≺ � N −1/2η−1/2 0 (1 + |λ|4) + (1 + |λ|)N −ε0/2� , λ ∈ R, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2) provided c ≤ V (f) ≤ C for some positive N-independent constants c and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Here the variance V (f) for a scaled test function f is defined by V (f) := 1 π2 � Ω0 � Ω′ 0 ∂ �f(ζ) ∂¯ζ ∂ �f(z) ∂¯z K(z, ζ)d¯ζdζd¯zdz, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3) where for z, ζ ∈ C/R the kernel K(z, ζ) is defined by K(z, ζ) := 2 β ∂ ∂ζ Tr �m′(z) m(z) � 1 − Sm(z)m(ζ) �−1 � + � 1 − 2 β � Tr [Sm′(z)m′(ζ)] + 1 2 ∂2 ∂z∂ζ � m(z)m(ζ), C(4)m(z)m(ζ) � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4) with C(4) denoting the matrix of fourth cumulants C(4) jk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The integration domains Ω0, Ω′ 0 in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3) are defined as Ω0 := {z ∈ C : | Im z| > N −ε0/2η0}, Ω′ 0 := {z ∈ C : | Im z| > 2N −ε0/2η0}, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5) and �f is the quasi-analytic extension of f, defined by �f(x + iη) = χ(η) (f(x) + iηf ′(x)) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6) where χ : R → [0, 1] is an even C∞ c (R) function supported on [−1, 1], satisfying χ(η) = 1 for |η| < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7 in [17]) Let E0, η0 satisfy the conditions of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Let f be the scaled test function with g ∈ C2 c (R) given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3), and let V (f) be the variance defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3), then V (f) = 1 2βπ2 ∥g∥2 ˙H1/2 + O � η0 log N + N −ε0� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7) 8 Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 implies that V (f) satisfies the condition of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1, hence φ′(λ) = −λφ(λ)V (f) + o (1) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='8) as N → ∞, for any fixed λ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' It then follows by L´evy’s continuity theorem that Tr f(H)−E [Tr f(H)] converges in distribution to a centered Gaussian with variance (2βπ2)−1 ∥g∥2 ˙H1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Therefore, to estab- lish Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2, it suffices to show that Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 hold, which is done in Sections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 and 7, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We restrict the proof to the real symmetric (β = 1) matrices for the sake of presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The complex Hermitian (β = 2) case differs solely in replacing the cumulant expansion formula (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6) with its complex analog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The obvious modifications are left to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 Characteristic Function of Linear Eigenvalue Statistics Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Using standard techniques of the characteristic function method imported from, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 of [17] (see also Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 of [19] and references therein), we can obtain the following series of estimates on the characteristic function of the linear eigenvalue statistics φ(λ) and its derivative φ′(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The proof is a relatively straightforward modification of similar arguments in [17], so we defer it to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Let φ(λ) be the characteristic function defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1), then, under the conditions of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2, the following estimates hold φ(λ) = E [�e(λ)] + O≺ � N −ε0/2� , φ′(λ) = i π � Ω0 ∂ �f ∂¯z E [�e(λ) {1 − E} [Tr G(z)]] d¯zdz + O≺ � |λ|N −ε0/2� , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9) where �e(λ) := exp �iλ π � Ω′ 0 ∂ �f ∂¯z {1 − E} [Tr G(z)] d¯zdz � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='10) Furthermore, for all z ∈ Dκ, we have E [�e(λ) {1 − E} [Tr G(z)]] = E [�e(λ) {1 − E} T (z, z)] + 2iλ π E � �e(λ) � Ω′ 0 ∂ �f ∂¯ζ ∂ ∂ζ T (z, ζ)d¯ζdζ � + iλ π E [�e(λ)] � Ω′ 0 ∂ �f ∂¯ζ Tr [Sm′(z)m′(ζ)] d¯ζdζ + iλ 2π E [�e(λ)] � Ω′ 0 ∂ �f ∂¯ζ ∂2 ∂z∂ζ � m(z)m(ζ), C(4)m(z)m(ζ) � d¯ζdζ + O≺ � (1 + |λ|4)(NΨ(z)Θ(z) + Ψ(z)η−1/2 0 ) � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='11) where the random function T (z, ζ) is defined as T (z, ζ) := Tr �m′(z) m(z) T (z, ζ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='12) We now proceed to estimate the first two terms on the right-hand side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='11) in such a way that E [�e(λ)] factors out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' By definition of the scaled test function (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3), the support of �f is contained inside a vertical strip centered at E0 of width ∼ η0, hence we limit the further analysis to the regime | Re ζ − Re z| ≲ η0 ≪ ǫ, where ǫ is defined in the statement of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We estimate the function T (z, ζ) using Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3 with weight matrix A := m′(z) m(z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' It follows from the bounds (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='14) that ∥A∥ℓ∞→ℓ∞ ≲ 1, hence for all z, ζ ∈ Dκ with Re z, Re ζ ∈ supp(f), T (z, ζ) = Tr �m′(z) m(z) � 1 − Sm(z)m(ζ) �−1� Sm(z)m(ζ) �2 � + E(z, ζ), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='13) 9 where the error term E(z, ζ) is analytic in both variables and admits the bound E(z, ζ) ≺ NΨ2(z)Ψ(ζ) + NΨ(z)Ψ2(ζ) + 1{Im z Im ζ<0}NΘ(z)Θ(ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='14) It follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='13) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='14) for ζ = z that E [�e(λ){1 − E} [T (z, z)]] ≺ NΨ(z)3, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='15) yielding the desired bound on the first term on the right-hand side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We now estimate the second term in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Fix z ∈ Dκ, and consider ζ that lie in Ω′ 0 defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Differentiating (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='13) with respect to ζ yields ∂ ∂ζ T (z, ζ) = ∂ ∂ζ Tr �m′(z) m(z) � 1 − Sm(z)m(ζ) �−1� Sm(z)m(ζ) �2 � + ∂ ∂ζ E(z, ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='16) To bound the derivative of the error term E(z, ζ), we use the following technical lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5 in [17]) Let K(z) be a holomorphic function on C\\R, then for all z ∈ C\\R and any p ∈ N, ���� ∂pK ∂zp (z) ���� ≤ Cp| Im z|−p sup |ζ−z|≤| Im z|/2 |K(ζ)|, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='17) where Cp > 0 is a constant depending only on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5 applied to the estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='14) implies that the error term ∂ζE(z, ζ) admits the bound ∂ ∂ζ E(z, ζ) ≺ N| Im ζ|−1� Ψ(z)2Ψ(ζ) + Ψ(z)Ψ(ζ)2 + Θ(z)Θ(ζ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='18) To proceed we require another technical lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4 in [19]) Let f be the scaled test function defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Let Ω be a domain of the form Ω := {z ∈ C : cN −τ ′η0 < | Im z| < 1, a < Re z < b}, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='19) such that supp(f) ⊂ (a, b) and τ ′, c are positive constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Let K(z) be a holomorphic function on Ω satisfying |K(z)| ≤ C| Im z|−s, z ∈ Ω, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='20) for some 0 ≤ s ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Then there exists a constant C′ > 0 depending only on g in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3), χ in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6), and s, such that ���� � Ω ∂ �f ∂¯z (x + iy)K(x + iy)dxdy ���� ≤ CC′η1−s 0 log N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='21) Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' It follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3) that ∥f∥1 ∼ η0, ∥f ′∥1 ∼ 1, ∥f ′′∥1 ∼ η−1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In case 1 ≤ s ≤ 2 the inequality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='21) follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4 in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' For 0 ≤ s < 1, the proof is conducted along the same lines, except the integration by parts is performed twice in the regime η0 ≤ | Im z| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6 and the matrix identity (1−X)−1X2 = (1−X)−1−X −1 yield the following expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' E � �e(λ) � Ω′ 0 ∂ �f ∂¯ζ ∂T ∂ζ d¯ζdζ � = E [�e(λ)] � Ω′ 0 ∂ �f ∂¯ζ ∂ ∂ζ Tr �m′(z) m(z) � 1 − Sm(z)m(ζ) �−1 � d¯ζdζ − E [�e(λ)] � Ω′ 0 ∂ �f ∂¯ζ Tr � Sm′(z)m′(ζ) � d¯ζdζ +O≺ � N 1/2Ψ(z)2η−1/2 0 + Ψ(z)η−1 0 + Θ(z)η−1 0 � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='22) Finally, from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='11) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='22), combined with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9) we conclude that φ′(λ) = −λV (f) E [�e(λ)] + �E(λ), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='23) 10 where V (f) is defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3), and �E(λ) is the total error term collected from previous derivations and integrated over d¯zdz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6 together with error estimates in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='11), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='15) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='18) provides the following bound on the error term �E = O≺ � N −1/2η−1/2 0 (1 + |λ|4) + |λ|N −ε0/2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='24) Under the conditions of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 V (f) is bounded, hence we conclude from the first estimate in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='23) that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' This concludes the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 6 Proof of the Local Laws for Two-point Functions In this section, we derive all the tools necessary to prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 and its specification for the two- point function T (z, ζ), Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' To make the notation more concise we introduce the convention G ≡ G(z), �G ≡ G(ζ), m ≡ m(z), �m ≡ m(ζ), �Ψ ≡ Ψ(ζ), Ψ ≡ Ψ(z), Θ ≡ Θ(z), �Θ ≡ Θ(ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' For a deterministic matrix W with entries |Wab| ≲ N −1, the quantity � a̸=y WaxGαa �Gaβ can be readily estimated in two special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' First, if each column of W is proportional to the vector of ones, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', Wab = wb depends only on b, then the summation over a yields wx([G �G]αβ − Gαy �Gyβ), and the estimate follows from the resolvent identity and the local laws in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Second, if the entries of X := (1 − Sm �m)−1W are bounded by CN −1, then one can obtain the estimate from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We show that these two special cases are exhaustive in the sense that any W can be represented as their linear combination with controlled coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' To this end, we prove that in the relevant regime, the operator (1 − Sm �m) has a very small destabilizing eigenvalue and an order one spectral gap above it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Moreover, if Π is the eigenprojector corresponding to the principal eigenvalue of (1 − Sm �m), then the ℓ∞ → ℓ∞-norm of the restriction of (1 − Sm �m)−1 to the kernel of Π is also an order one quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Finally, we show that the vector of ones 1 is sufficiently separated from the kernel of Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 Stable Direction Local Law For any N × N deterministic matrix W, and any indices x, y, α, β, we define the quantities Fxy αβ(W) := � a̸=y WaxGαa �Gaβ, f xy α (W) := mα �mα([(1 − Sm �m)−1W]αx − δαyWαx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1) We prove the following estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' For any z, ζ ∈ Dκ and any deterministic N × N matrix X, Fxy αβ((1 − Sm �m)X) = δαβf xy α ((1 − Sm �m)X) + O≺ � N ∥X∥max Ψ�Ψ(Ψ + �Ψ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2) provided ∥X∥max := max j,k |Xjk| ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We use the following self-improving mechanism for stochastic domination bounds, borrowed, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', from [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3 in [14]) Let X be a random variable such that 0 ≤ X ≺ N C for some C > 0, and let Ξ ≥ 0 be a deterministic quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Suppose there exists a constant q ∈ [0, 1), such that for any Φ satisfying Ξ ≤ Φ ≤ N C, and any d ∈ N, we have the implication X ≺ Φ =⇒ E � |X|2d� ≺ 2d � k=1 � ΦqΞ1−q)k E � |X|2d−k� , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3) then X ≺ Ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Let Y := (1 − Sm �m)X, then the quantity we need to estimate is [GY ]yx = Fxy yy (Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' It follows from the local law in the form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4) that Fxy αβ(Y ) ≺ N ∥X∥max Ψ�Ψ =: Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4) 11 Let Φ be a deterministic control parameter admitting the bounds (Ψ + �Ψ)Λ ≤ Φ ≤ Λ, such that Fxy αβ(Y ) − δαβf xy α (Y ) ≺ Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5) It follows trivially from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5) that Fxy αβ(Y ) ≺ Φ + δαβΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6) Let ∂jk denote the partial derivative with respect to the matrix element Hjk, then the partial derivatives of Fxy αβ are given by ∂abFxy αβ(Y ) = −(1 + δab)−1(GαaFxy bβ (Y ) + GαbFxy aβ(Y ) + Fxy αb (Y ) �Gaβ + Fxy αa(Y ) �Gbβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7) We combine the vector Dyson equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4) and the resolvent identity zG = HG − 1 to obtain �Gaβ = − �ma � b � Hab �Gbβ + Sab �mb �Gaβ � + �maδaβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='8) Let d ∈ N, define P ≡ P(d − 1, d) := (Fxy αβ(Y ) − δαβf xy α (Y ))d−1(Fxy αβ(Y ) − δαβf xy α (Y ))d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' For any p ∈ N, define Mp := E � |Fxy αβ(Y ) − δαβf xy α (Y )|p� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Plugging (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='8) into the definition (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1) and applying the cumulant expansion formula of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6, we obtain E � Fxy αβ(X)P � = � a̸=y ma �maXax E � Fay αβ(S)P � + δαβf xy α (Y ) E[P] + δαβδβySyym2 y �m2 yXyx E[P] (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9a) + E �� a̸=y � b �maXaxSab � Gαa( �Gbb − �mb) �Gaβ + Gαb(Gaa − ma) �Gbβ �P � (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9b) + E �� a̸=y � b̸=a �maXaxSabGαa � Gba + �Gba � �GbβP � + R2 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9c) + � a̸=y Xaxma �maSay E �� Gαy �Gyβ − δαyδyβmy �my �P � (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9d) + δβ̸=y �mβXβx E � (Gαβ − δαβmβ)P � − E �� a̸=y �maXaxGαa � b Sab �Gbβ∂abP � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9e) where R2 is the total error coming from the higher order cumulants, and all unrestricted summations are from 1 to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We successively bound the terms (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9b)-(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9e) appearing on the right-hand side of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' By condition (A), local law (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4), upper bound (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2), and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5), it follows that the terms (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9b) and the first term in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9c) are bounded by O≺((Ψ + �Ψ)ΛM2d−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Similarly, the term (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9d) and the first term in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9e) are bounded by O≺(∥X∥max (Ψ + �Ψ)M2d−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We bound the second term in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' It follows by (A), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4), bounds (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6), and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7) that � b Sab �Gbβ∂abP ≺ (Ψ + �Ψ + δαa + δaβ)�ΨΦM2d−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='10) Hence, the second term in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9e) is bounded by O≺ � (Ψ + �Ψ)ΛΦM2d−2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Finally, it is easy to check using estimates (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='16), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6) and identity (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7), together with condition (A) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2), that the error term R2 ≺ (Ψ + �Ψ)ΛM2d−1 + (Ψ + �Ψ)ΛΦM2d−2 + (Ψ + �Ψ)ΛΦ2M2d−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Observe that the first term on the right-hand side of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9a) can be expressed as � a̸=y ma �maXax E � Fay αβ(S)P � = E � Fay αβ(X)P � − E � Fay αβ(Y )P � − my �myXyx E � Fyy αβ(S)P � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='11) where the last term is bounded by O≺(N −1ΛM2d−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Combining (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='11) yields E � |Fxy αβ(Y ) − δαβf xy α (Y )|2d� ≺ � Ψ + �Ψ � ΛΦ2M2d−3, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='12) for any control parameter Φαβ,y satisfying (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Hence, by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2, Fxy αβ(Y ) = δαβf xy α (Y ) + O≺ � Λ(Ψ + �Ψ) � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='13) which concludes the proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 12 Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' If z and ζ are in the same (upper or lower) half-plane, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 implies Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Indeed, the bound (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='12) in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4 shows that provided η�η > 0, X := (1−Sm �m)−1W satisfies |Xjk| ≲ N −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Applying Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 to X = (1 − Sm �m)−1W then yields (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6), and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7) follows by summing (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We turn to the case of z and ζ lying in different (upper and lower) half-planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 Stability Operator Analysis In this subsection we obtain all the properties of the stability operator (1 − Sm(z)m(ζ)) that we use in combination with Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 to finish the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 for z, ζ lying in opposite half-planes, as outlined in the beginning of Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' For two spectral parameters z, ζ, let η := Im z, and �η := Im ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Without loss of generality, we assume in the following that Re z ∈ Iκ, η > 0 and Re ζ ∈ Iκ, �η < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' For the remainder of this subsection, we use the following notation F ≡ F(z) := |m(z)|S|m(z)|, B ≡ B(z, ζ) := 1 − Sm(z)m(ζ), B0 ≡ B0(z) := 1 − S|m(z)|2 = |m(z)|−1(1 − F)|m(z)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='14) We view the operator B as a perturbation of B0 = B(z, ¯z), since |ζ − ¯z| is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We deduce the desired properties of B from those of B0, which, in turn, follow from the lower bound on the spectral gap of F found in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Let {ψj}N j=1 denote the eigenvalues of F (with multiplicity) in descending order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Then, by Per- ron–Frobenius theorem, the principal eigenvalue ψ1 is real, and it coincides with the spectral radius ∥F∥ℓ2→ℓ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Furthermore, by taking the imaginary part of the vector Dyson equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4) and multi- plying both sides by |m| coordinate-wise, we obtain � 1 − F �Im m |m| = η|m|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='15) Furthermore, by condition (A), for every j we have (S Im m)j ∼ ⟨Im m(z)⟩ ∼ ρ(z), where ρ(z) is the harmonic extension of the self-consistent density of states ρ(x) defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3) into C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Hence by taking the imaginary part of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4), we get Im mj |mj| ∼ |mj|(ρ(z) + η), , j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='16) Therefore, by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='15) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='16), 1 − ψ1 ≲ η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Together with an upper bound (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='10) on ∥F∥ℓ2→ℓ2, this implies that 1 − ψ1 ∼ η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' It follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9) that the principal eigenvalue of F is separated from the rest of the spectrum by an annulus, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', there exist r > 0 and δ > 0 independent of z and N such that |1 − ψ1| < r − δ, and |1 − ψj| > r + δ, j ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='17) In the remainder of this subsection, we show that for all ζ sufficiently close to ¯z, the eigenvalue of B with the smallest modulus is also separated from the rest of the spectrum by an annulus of order one width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Using the argument principle and Jacobi’s formula, one can express the number of eigenvalues (with multiplicity) of a matrix X inside a domain Ω by a contour integral NX(Ω) = 1 2πi � ∂Ω Tr(w − X)−1dw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='18) To show the eigenvalue separation for B, we begin by estimating the norm of the resolvent of B inside the annulus Ar,δ := {w ∈ C : r − 3δ/4 ≤ |w| ≤ r + 3δ/4}, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='19) with r and δ as in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 13 Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' There exists ε1 > 0 and �C > 0 independent of N and z such that ���(w − B(z, ζ))−1��� ≤ �C (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='20) holds for all w ∈ Ar,δ and all ζ such that Re ζ ∈ Iκ, Im ζ < 0 and |ζ − ¯z| ≤ ε1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (The norm ∥·∥ is induced by either ℓ2 or ℓ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=') Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Observe that ��(w − B)−1�� ≤ ��� � 1 − (w − B0)−1(B − B0) �−1��� ��(w − B0)−1��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Since (w − B0)−1 = −|m|−1(1 − w − F)−1|m| and |m| ∼ 1, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='17) implies that ��(w − B0)−1�� ≤ C min j |ψj − w| ≤ 4C δ , w ∈ Ar,δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='21) From the uniform bounds (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='14) on |m| and |m′| we have ∥B − B0∥ ≲ |ζ − ¯z|, which implies that there exists ε1 > 0 such that ∀ζ : |ζ − ¯z| ≤ ε1, ∥B − B0∥ ≤ δ 8C , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='22) where C is the constant in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' It follows immediately that ��� � 1 − (w − B0)−1(B − B0) �−1��� ≤ 2 and hence ��(w − B)−1�� ≤ 8C δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='23) Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4 implies that for any sufficiently large fixed N the integrand in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='18) with X := B is uniformly bounded in Ω := Ar,δ for all ζ such that |ζ − ¯z| ≤ ε1, hence by analyticity NB(z,ζ)(Ar,δ) = 0, |ζ − ¯z| ≤ ε1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='24) Since the eigenvalues of B(z, ζ) are continuous in ζ, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='24) implies that no eigenvalue can move between the two connected components of C\\Ar,δ, which together with (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='17) yields the following claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' For any sufficiently large N, the equalities NB({|w| < r − 3δ/4}) = NB0({|w| < r − 3δ/4}) = 1, NB({|w| > r + 3δ/4}) = NB0({|w| > r + 3δ/4}) = N − 1, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='25) hold for any ζ such that Re ζ ∈ Iκ, Im ζ < 0 and |ζ − ¯z| ≤ ε1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5 now allows us to define the principal eigenprojector Π of B as a contour integral Π ≡ Π(z, ζ) := 1 2πi � |ξ|=r (ξ − B(z, ζ))−1dξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='26) Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5 asserts that the contour {|ξ| = r} encircles exactly one eigenvalue of B with multiplicity, hence Π is a rank one eigenprojector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We now prove that the restriction of B−1 to the range of (1 − Π) is bounded by a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' For all z, ζ such that Re z, Re ζ ∈ Iκ, Im z Im ζ < 0 and |ζ − ¯z| ≤ ε1, ��B−1(1 − Π) �� ℓ∞→ℓ∞ ≤ �c, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='27) where �c depends only on the constants in conditions (A), (B) and κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' By expression (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='26) for Π we have B−1(1 − Π) = − 1 2πi � |ξ|=r 1 ξ (ξ − B)−1dξ (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='28) 14 Hence the norm of B−1(1 − Π) is bounded by ��B−1(1 − Π) �� ℓ∞→ℓ∞ ≤ 1 2π 2π � 0 ��� � reiθ − B �−1��� ℓ∞→ℓ∞ dθ ≤ 8C δ , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='29) using the bound in Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4 on the circle {|ξ| = r} which lies inside Ar,δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Finally, we show that the vector of ones is sufficiently separated from the kernel of Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' This ensures a stable decomposition of the space into the direct sum of the range of (1 − Π) and the span of 1, so we can apply the local laws to each of the components separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' There exists ε > 0 independent of N and z such that for all ζ with Re ζ ∈ Iκ, Im ζ < 0 and |ζ − ¯z| ≤ ε, ∥Π1∥∞ ∥Π∥ℓ∞→ℓ∞ ≥ c, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='30) where c > 0 is a constant independent of N and z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Define the projector Π0 corresponding to B0 via (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Then Π0 = |m|−1�Π0|m|, where �Π0 is the orthoprojector corresponding to the principal eigenvalue of the Hermitian operator F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Since |m| ∼ 1 we have ∥Π0∥ℓ∞→ℓ∞ ≤ C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Moreover, by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3, the ℓ2-normalized eigenvector v corresponding to the principal eigenvalue of F has entries vj ≥ 0 with vj ∼ N −1/2, hence ∥Π01∥∞ = ���|m|−1�Π0|m|1 ��� ∞ = ��|m|−1v �� ∞ ⟨v, |m|⟩ ≥ c0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='31) where c0 > 0 is a constant independent of N and z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Similarly to the proof of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='22), for any γ ∈ (0, 1] there exists εγ > 0, such that the bound ∥B − B0∥ℓ∞→ℓ∞ ≤ γ δ 8C (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='32) holds for all ζ ∈ D− κ with |ζ − ¯z| ≤ εγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Here δ is defined in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='17) and C > 0 is the constant in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We choose εγ to be smaller than ε1 of Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4, then for all ζ with Re ζ ∈ Iκ, Im ζ < 0 such that |ζ − ¯z| ≤ εγ we have ∥Π − Π0∥ℓ∞→ℓ∞ ≤ r 2π 2π � 0 ��(reiθ − B)−1 − (reiθ − B0)−1�� ℓ∞→ℓ∞ dθ ≤ r 2π 2π � 0 ��(reiθ − B)−1(B − B0)(reiθ − B0)−1�� ℓ∞→ℓ∞ dθ ≤ r · 8C δ · γ δ 8C · 4C δ = γ 4Cr δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='33) Here we used inequalities (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='21) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='23) in the second to last step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We set the value of γ to be γ0 := min � 1, c0δ 8Cr � , which guarantees that ∥Π1∥∞ ≥ ��∥Π01∥∞ − ∥Π − Π0∥ℓ∞→ℓ∞ ∥1∥∞ �� ≥ c0 − γ0 4Cr δ ≥ c0 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='34) Finally, observe that ∥Π∥ℓ∞→ℓ∞ ≤ ∥Π0∥ℓ∞→ℓ∞ + ∥Π − Π0∥ℓ∞→ℓ∞ ≤ C0 + c0/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='35) This proves the claim with c := c0/(2C0 + c0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 15 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3 Finishing the Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Recall that the objective is to estimate the quantities defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In- stead of estimating � a̸=y waGαa �Gaβ directly, it is more convenient to work with objects of the type � a̸=y WaxGαa �Gaβ, since they generalize quantities appearing in both (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The redundant index x can be eliminated by setting Wax := wa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In the case Im z Im ζ > 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7) follow immediately from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='12) and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 (see Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Therefore, we focus on the case Im z Im ζ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Since Π has rank one and Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7 asserts that Π1 ̸= 0, the kernel of Π together with 1 span CN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Therefore we can decompose each column of the matrix W into a linear combination of 1 and an element of ker Π, that is, there exists an N × N matrix Y and a vector s ∈ CN such that W = Y + 1s∗, ΠY = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='36) We multiply the first equality in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='36) by Π from the left, apply both sides to the a-th standard basis vector ea of CN and take the ℓ∞-norm to deduce ∥ΠWea∥∞ = |sa| ∥Π1∥∞ , a ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='37) By assumption, ∥W∥max ≲ N −1, hence ∥Wea∥∞ ≲ N −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Using Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7 we get |sa| ≲ N −1 ∥Π∥ℓ∞→ℓ∞ ∥Π1∥∞ ≲ N −1, a ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='38) We combine (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='36) and the resolvent identity in the form (z − ζ)G �G = G − �G to obtain � a̸=y WaxGαa �Gaβ = � a̸=y YaxGαa �Gaβ + gy αβ¯sx, gy αβ := Gαβ − �Gαβ z − ζ − Gαy �Gyβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='39) Define the N × N matrix X := (1 − Sm �m)−1 Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' It follows from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='36) that Y = (1 − Π)Y , hence X = (1 − Sm �m)−1(1 − Π)Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Furthermore, estimates ∥W∥max ≲ N −1, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='36), and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='38) imply that |Yab| ≲ N −1 for all a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Since by Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6 ��(1 − Sm �m)−1(1 − Π) �� ℓ∞→ℓ∞ ≲ 1, we conclude that ∥X∥max = max a,b |Xab| ≲ N −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='40) First, using (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='40), we can apply Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 to the first term in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='39) to obtain � a̸=y YaxGαa �Gaβ = δαβmα �mα([(1 − Sm �m)−1Y ]αx − δαyYαx) + O≺ � Ψ2 �Ψ + Ψ�Ψ2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='41) Using (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='36), we proceed by computing mα �mα[(1 − Sm �m)−1Y ]αx = � m �m � 1 − Sm �m �−1 (W − 1s∗) � αx = � m �m � 1 − Sm �m �−1W � αx − � m �m � 1 − Sm �m �−11 � α¯sx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='42) Finally, it follows from subtracting the vector Dyson equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4) for z and ζ that m �m � 1 − Sm �m �−11 = m − �m z − ζ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='43) Next, we estimate the second term in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Applying the local law in the form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4), we obtain gy αβ = δαβ mα − �mα z − ζ − δαβδαymα �mα + O≺ � (|η| + |�η|)−1(Ψ + �Ψ) � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='44) where we used that |z − ζ| ≥ |η| + |�η|, since η�η < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Combining (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='38), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='39), and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='41)-(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='44) yields � a̸=y WaxGαa �Gaβ = δαβ � m �m � 1 − Sm �m �−1W � αx − δαβδαy[m �mW]αx + O≺ � (Ψ + �Ψ)(Ψ�Ψ + min{Θ, �Θ}) � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='45) 16 which proves (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6) by setting Wax := wa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' To prove (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7), we observe that by setting x = y = α = β = b in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='39) and summing over b yields � b � a̸=b WabGba �Gab = � b � a̸=b YabGaa �Gab+⟨s, g⟩, gb := Gbb − �Gbb z − ζ −Gbb �Gbb, b ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='46) To estimate ⟨s, g⟩, we use (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='38) and the averaged local law (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3) to obtain � s, g � = � s, m − �m z − ζ − m �m � + O≺ � (|η| + |�η|)−1(Θ + �Θ) � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='47) where we used that |z − ζ| ≥ |η| + |�η|, since η�η < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Setting x = y = α = β = b in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='41), summing over b, using the identities (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='42) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='43), and combining the result with (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='47), we deduce that � b � a̸=b WabGba �Gab = Tr � m �mSm �m � 1 − Sm �m �−1W � + NO≺ � Ψ�Ψ(Ψ + �Ψ) + Θ�Θ � , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='48) where we used that (|η| + |�η|)−1(Θ + �Θ) = NΘ�Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' This establishes (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7) and concludes the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We outline the steps needed to achieve the optimal error estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' First, one needs to adapt the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' More specifically, replace the decomposition (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='36) with W = Y + 1s∗ + q1∗, such that Π(z, ζ)Y = Y Πt(ζ, z) = 0, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='49) where Π(z, ζ) is the destabilizing eigenprojector defined in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The terms involving s and q are handled using the averaged local law (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3), similarly to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' For the remaining term, R := � y Fyy yy , we adapt the mechanism of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 by using the following iterative scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In the first step, we apply an expansion similar to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9) to the partial derivative ∂jkR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' This improves the error in the estimate on R by a factor of (Ψ + �Ψ)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' If we expand ∂lp∂jkR in a similar manner, we gain another (Ψ + �Ψ)1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Iterating this approach we can estimate R with an error stochastically dominated by NΨ�Ψ(Ψ + �Ψ)2−2−d for any given integer d (where d is the maximal order of expanded partial derivatives).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' By Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1, this is sufficient to establish (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Similar arguments in the context of random band matrices can be found in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Proof of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9) on Txy(ζ, z) follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6) by setting α = β = y and wa := Sxa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='10) on Tr[AT (z, ζ)] follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7) by setting W := SAt, which satisfies |Wab| ≲ N −1 ∥A∥ℓ∞→ℓ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' This concludes the proof of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Note that estimates (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7) (also with the improved error term (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='12)) hold without omission of indices in the a summation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Indeed, it follows from Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 that � a waGαa �Gaβ = δαβ � m �m � 1 − Sm �m �−1w � α + O≺ � (Ψ + �Ψ)(Ψ�Ψ + 1{η�η<0} min{Θ, �Θ}) � , � a,b WabGba �Gab = Tr � m �m � 1 − Sm �m �−1W � + O≺ � N(Ψ + �Ψ)Ψ�Ψ + 1{η�η<0}NΘ�Θ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='50) 7 Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 In this section, we compute the variance V (f) defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3) for mesoscopic C2 c test functions f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In [17], the limiting variance was computed for several types of C∞ test functions, including compactly supported ones;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' however, V (f) is computed with an O(1) error (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7 in [17]), which is not negligible in the setting of the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' To obtain effective error bounds, we augment the proof laid out in [17] by performing further integration by parts in the integral representation of V (f), thus eliminating the f ′ terms, improving the error by a factor of O(η0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Throughout this section, we adhere to the notation m ≡ m(z), �m ≡ m(ζ), η := Im z, �η := Im ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 17 The stability operator (1 − Sm �m) can be expressed in terms of the self-saturated energy operator F, defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6), via the following identity 1 − Sm �m = |m �m|−1/2 (U∗ − F(z, ζ)) |m �m|1/2U, U := m �m |m �m|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1) Furthermore, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9), the operator F can be decomposed such that F(z, ζ) = ψ1(z, ζ) v(z, ζ) � v(z, ζ) �∗ + A(z, ζ), A(z, ζ)v(z, ζ) = 0, ∥A(z, ζ)∥ℓ2→ℓ2 ≤ 1 − �δ, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2) where ψ1, v is the principal eigenvalue-eigenvector pair of F, and �δ is the constant in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Let R ≡ R(z, ζ) denote (U∗(z, ζ) − A(z, ζ))−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In the sequel, we drop the arguments and write A ≡ A(z, ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Lower bound (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='8) and the inequality in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2) imply that ∥R∥ℓ2→ℓ2 + ∥R∥ℓ∞→ℓ∞ ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3) In the following lemma, we collect the perturbative estimates on the saturated self-energy operator F and related quantities established in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='52), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='60), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='71), and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='67) in [17]) Let w, ζ1, ζ2 be spectral parameters in Iκ + i[−1, 1], and let F be the operator defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6), then the principal eigenvalue- eigenvector pair ψ1, v of F satisfies ∥v(w, ζ1) − v(w, ζ2)∥ℓ2→ℓ2 + |ψ1(w, ζ1) − ψ1(w, ζ2)| ≲ |ζ1 − ζ2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4) Furthermore, for operator A defined in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2), we have the estimate ∥F(w, ζ1) − F(w, ζ2)∥ℓ2→ℓ2 + ∥A(w, ζ1) − A(w, ζ2)∥ℓ2→ℓ2 ≲ |ζ1 − ζ2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5) Let z := x + iη, ζ := y − iη, with x, y ∈ Iκ, 0 ≤ η ≤ 1, then ψ1 � v, Rm′ m U∗Rv � = ψ1(z, z) � v(z, z)m′ m v(z, z) � + O(|x − y|) (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6) Let ω ≡ ω(z, ζ) := 1 − ψ1⟨v, Rv⟩, then ω(z, ζ) = 1 − ψ1(z, z) + ψ1(z, z)(x − y) � v(z, z)m′ m v(z, z) � + O(|x − y|2), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7) Moreover, there exists ε > 0 independent of N, such that for all x, y ∈ Iκ satisfying |x − y| ≤ ε, |ω(z, ζ)| ≳ η + |x − y|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='8) Finally, for z := x + iη with x ∈ Iκ, the following identity holds lim η→+0 � v(z, z)m′ m v(z, z) � = iπ 2 ρ(x) ���� Im m(x + i0) |m(x)| ���� −2 2 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9) By our choice of κ, E0 is in the interior of the bulk interval Iκ, defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5) , hence if we define ˆε := min{ε/4, dist(E0, R\\Iκ)}, then ˆε ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Furthermore, since the function g is compactly supported, we assume that supp(f) ⊂ [E0 − ˆε, E0 + ˆε] for large N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Let η∗ ≡ η∗(N) satisfy 0 < η∗ ≤ N −100, then V (f), defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3), admits the estimate V (f) = 1 4π2 �� [E0−ˆε,E0+ˆε]2 (f(y) − f(x))2 �K(x + iη∗, y − iη∗)dxdy + O � η0 + N −ε0� , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='10) where �K(z, ζ) := −2 Re Tr �m′ m (1 − Sm �m)−1Sm �m′(1 − Sm �m)−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='11) 18 In preparation for the proof of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 we define an auxiliary function L(z, ζ) L(z, ζ) := Llog(z, ζ) + L1(z, ζ), Llog(z, ζ) := −2 log det {1 − Sm �m} , L1(z, ζ) := − Tr [Sm �m] + 1 2 � m �m, C(4)m �m � , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='12) where log is the principal branch of the complex logarithm, and C(4) is the matrix of the fourth cumu- lants of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' By Jacobi’s formula for the derivative of the determinant, it follows from the definitions of L and K, that for all z, ζ ∈ C\\R ∂2 ∂ζ∂z L(z, ζ) = K(z, ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='13) Furthermore, by condition (A) and the upper bound (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2), it follows that |Llog(z, ζ)| ≤π + log |det {1 − Sm �m}| ≲ 1 + Tr � (1 − Sm �m)∗ (1 − Sm �m) − I � ≲ 1, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='14) where in the last line we used � (1 − Sm �m)∗ (1 − Sm �m) − I � jj ≲ N −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The partial derivatives of L1 contribute only sub-leading terms to L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Indeed, we have the estimates L1(z, ζ) ≲ 1, ∂ ∂zL1(z, ζ) ≲ 1, ∂2 ∂ζ∂z L1(z, ζ) ≲ 1, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='15) where we used the moment condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2) to bound Sjk and C(4) jk , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2) to get the upper bound m, �m ≲ 1, and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='14) to obtain m′, �m′ ≲ 1, since [E0 + ˆε, E0 − ˆε] ⊂ Iκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The following claim collects the bounds on K and ∂zL that together with (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='14) enable integration by parts in the definition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3) of the variance V (f), which is the essence of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Claim 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 and Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6 in [17]) Let K(z, ζ) and L(z, ζ) be as defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4) (with β = 1) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='12) respectively, then for all z, ζ ∈ C\\R with Re z, Re ζ ∈ [E0 − ˆε, E0 + ˆε] and | Im z|, | Im ζ| ≤ 1 we have K(z, ζ) ≲ 1 + 1{η�η<0}(|η| + |�η|)−2, ∂ ∂z L(z, ζ) ≲ 1 + (| Re z − Re ζ| + |η| + |�η|)−1, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='16) where η := Im z, �η := Im ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Proof of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Define Ω∗ := {z ∈ C : 1 > | Im z| > η∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Recall the definition of V (f) from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' First, we prove that V (f) = 1 π2 � Ω∗ � Ω∗ ∂ �f(ζ) ∂¯ζ ∂ �f(z) ∂¯z K(z, ζ)d¯ζdζd¯zdz + O � N −ε0� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='17) It follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6) that ∂ �f ∂¯z = 1 2 � −ηχ′(η)f ′(x) + i � ηχ(η)f ′′(x) + χ′(η)f(x) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='18) Moreover, for all z with | Im z| < 1/2, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='18) and the properties of χ in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6) imply ∂ �f ∂¯z = i Im z 2 f ′′(Re z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='19) Let V∗(f) denote the integral on right hand side of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='17), and define η1 := N −ε0/2η0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' It follows from the first inequality in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='16), and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='19) that |V (f) − V∗(f)| ≲ �� R2 |f ′′(x)f ′′(y)| dxdy η1 � η∗ 2η1 � η∗ η�η (η + �η)2 d�ηdη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='20) 19 Note that η�η ≤ (η + �η)2/4, hence the integral over d�ηdη is bounded by η2 1/2, and since ∥f ′′∥1 ∼ η−1 0 , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='17) is established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We write z := x + iη, ζ := y + i�η and plug (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='13) into the expression (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='17) for V (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Using the fact that ∂zu = −i∂ηu for any holomorphic function u(z), and integrating by parts in η, we obtain V (f) = i π2 �� R2 dxdy � |�η|>η∗ ∂ �f(ζ) ∂¯ζ � |η|>η∗ ∂2 �f(z) ∂η∂¯z ∂ ∂ζ L(z, ζ)d�ηdη − i π2 �� R2 dxdy � |�η|>η∗ ∂ �f(ζ) ∂¯ζ � η=±η∗ ∂ �f ∂¯z (x + iη) ∂ ∂ζ L(z, ζ)d�η + O � N −ε0� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='21) The second estimate in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='16), expression (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='18) and the estimates ∥f ′′∥1 ∼ η−1 0 , ∥f ′∥1 ∼ 1, ∥f∥1 ∼ η0 imply that the boundary term in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='21) is dominated by O≺(η∗η−2 0 ), which is smaller than O (N −ε0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Similarly, integrating the first term on the right hand side of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='21) by parts in �η we get V (f) = − 1 π2 � Ω∗ � Ω∗ ∂2 �f(z) ∂¯z∂η ∂2 �f(ζ) ∂¯ζ∂�η L(z, ζ)d¯ζdζd¯zdz + 1 π2 �� R2 dxdy � |η|>η∗ ∂2 �f(z) ∂η∂¯z � �η=±η∗ ∂ �f ∂¯ζ (y + i�η)L(z, y + i�η)dη + O � N −ε0� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='22) It follows from (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='14) and the expression (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='18) that the boundary term (the second line of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='22)) is again dominated by O≺(N −ε0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We apply Stokes’ theorem to (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='22) twice: once in z and once in ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Considering that ∂η �f(z) vanishes on the boundary of Ω∗ except for the lines {Im z = ±η∗}, this results in V (f) = 1 4π2 �� R2 � η,�η=±η∗ sign (η�η) ∂ �f(x + iη) ∂η ∂ �f(y + i�η) ∂�η L(x + iη, y + i�η)dxdy + O � N −ε0� = − 1 2π2 �� R2 f ′(x)f ′(y) �L(x, y)dxdy + O � N −ε0� , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='23) where �L(x, y) := Re [L(x + iη∗, y + iη∗) − L(x + iη∗, y − iη∗)] (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='24) We restrict the integrations in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='23) to [E0 − ˆε, E0 + ˆε], since this interval contains the support of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Furthermore, for all y ∈ supp(f), y − E0 ≲ η0, hence |y − E0 ± ˆε| ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' By symmetry of L(z, ζ), and the second estimate in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='16) it follows that ∂ ∂y �L(E0 ± ˆε, y) ≲ 1, y ∈ supp(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='25) We write f ′(y) = ∂y (f(y) − f(x)), perform integration by parts in y and integrate the boundary term by parts in x to obtain V (f) = 1 2π2 E0+ˆε � E0−ˆε E0+ˆε � E0−ˆε f ′(x) (f(y) − f(x)) ∂ ∂y �L(x, y)dxdy + 1 4π2 E0+��ε � E0−ˆε (f(x))2 ∂ ∂x � �L(x, E0 + ˆε) − �L(x, E0 − ˆε) � dx + O � N −ε0� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='26) Since ∥f∥2 2 ≲ η0, it follows from (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='25) that the second integral in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='26) is O (η0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Similarly, integrating (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='26) by parts in x and using (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='26) to substitute one of the emerging itegrals for −V (f) + O (N −ε0 + η0), we get 2V (f) = 1 2π2 E0+ˆε � E0−ˆε E0+ˆε � E0−ˆε (f(y) − f(x))2 ∂2 ∂x∂y �L(x, y)dxdy + O � η0 + N −ε0� , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='27) 20 where we again used (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='25) to estimate the boundary term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' For any holomorphic function u(z) of z = x + iη, we have ∂xu = Re[∂zu], hence ∂x∂y �L(x, y) = Re [K(x + iη∗, y + iη∗) − K(x + iη∗, y − iη∗)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Finally, in view of in view of the first estimate in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='16), ∂z∂ζLlog(x + iη∗, y + iη∗) ≲ 1, so its contribution is also bounded by O≺(η0 ∥g∥2 2 + η2 0 ∥g∥2 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Moreover, it follows from the last estimate in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='15) that we can replace K(x+iη∗, y −iη∗) by ∂z∂ζLlog(x+iη∗, y −iη∗), since the contribution of the remaining terms is bounded by O≺(η0 ∥g∥2 2 + η2 0 ∥g∥2 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' This concludes the proof of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Once Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 is established, we can follow the method of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7 in [17] to finish the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Fix x, y ∈ [E0 − ˆε, E0 + ˆε] and write z := x + iη∗, ζ := y − iη∗, as in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' It follows from (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2) that the kernel �K(z, ζ) can be written as �K(z, ζ) = −2 Re Tr �m′ m U∗� R + ψ1 ω Rvv∗R � F �m′ �m � R + ψ1 ω Rvv∗R �� , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='28) where ω is defined in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Expanding the brackets in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='28), collecting like terms according to the powers of ω−1, and using the cyclic property of trace yields �K(z, ζ) = −2 Re �ψ2 1 ω2 � v, Rm′ m U∗Rv �� v, RF �m′ �m Rv �� + O � 1 + ω−1� , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='29) since Tr � m′ m U∗RF � m′ � m R � , Tr � m′ m U∗RF � m′ � m Rvv∗R � , and Tr � m′ m U∗Rvv∗RF � m′ � m R � are all O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The first scalar product in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='29) can be estimated using (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' We compute the second scalar product in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' It follows from uniform bounds (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='14) that ∥m(z)− m(¯ζ)∥∞ ≲ |x− y|, and hence ∥U(z, ζ) − 1∥ℓ2→ℓ2 ≲ |x− y|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Together with estimates (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4), this yields ψ1 � v, RF �m′ �m Rv � = ⟨v(ζ, ζ), F(ζ, ζ) � m′ �m v(ζ, ζ) � + O(|x − y|), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='30) where we used the identity R(¯ζ, ζ)v(ζ, ζ) = (1 − A(ζ, ζ))−1v(ζ, ζ) = v(ζ, ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' It follows from the estimate on v in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4) that ∥v(ζ, ζ) − v(y, y)∥2 ≲ η∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Vector v(y, y) is the ℓ2- normalization of |m(y)|−1 Im m(y + i0), hence it satisfies F(y, y)v(y, y) = v(y, y) by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Therefore using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='14) and the lower bound in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5), we obtain ∥F(ζ, ζ)v(ζ, ζ) − v(ζ, ζ)∥2 ≲ η∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='31) Substituting (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='31) into (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='30) yields ψ1 � v, RF �m′ �m Rv � = ⟨v(ζ, ζ), �m′ �m v(ζ, ζ) � + O(|x − y| + η∗), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='32) Combining (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='28) with estimates (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='8) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='32) yield �K(z, ζ) = −2 Re �ψ1(z, z)ψ1(ζ, ζ) ω2 � v(z, z)m′ m v(z, z) � ⟨v(ζ, ζ), �m′ �m v(ζ, ζ) �� + O(1 + ω−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='33) It follows by (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7) that lim η∗→+0 �K(x + iη∗, y − iη∗) = 2|x − y|−2 + O(|x − y|−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='34) Since f ∈ C2 c (R), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='33) implies that the integrand in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='10) is uniformly bounded in η∗ ∈ [0, N −100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Therefore, we can take the limit η∗ → 0 in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='10), and apply the boundary estimate (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='34) to obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' V (f) = 1 2π2 �� [E0−ˆε,E0+ˆε]2 (f(x) − f(y))2 (x − y)2 dxdy + O � η0 log N + N −ε0� , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='35) because the contribution of O(|x − y|−1) to the integral (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='10) is bounded by O(η0 log N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 21 Finally, the contribution of the regime (x, y) /∈ [E0 − ˆε, E0 + ˆε]2 to the integral �� R2 (f(x) − f(y))2 (x − y)2 dxdy = ∥f∥2 ˙H1/2 = ∥g∥2 ˙H1/2 , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='36) is bounded by O≺(η0), therefore V (f) = 1 2π2 ∥g∥2 ˙H1/2 + O � η0 log N + N −ε0� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='37) This concludes the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Appendix A Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4 We use the Helffer–Sj¨ostrand representation to express the linear eigenvalue statistics in terms of the resolvent of H (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 in [19] for references), {1 − E} [Tr f(H)] = 1 2π � C ∂ �f ∂¯z {1 − E} [Tr G(z)] d¯zdz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1) The characteristic function φ then admits the form φ(λ) = E [e(λ)] , e(λ) := exp � iλ 1 2π � C ∂ �f ∂¯z {1 − E} [Tr G(z)] d¯zdz � , λ ∈ R, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2) and its derivative φ′ is given by φ′(λ) = E � e(λ) i 2π � C ∂ �f ∂¯z {1 − E} [Tr G(z)] d¯zdz � , λ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3) As observed in [19], the regime | Im z| ≤ N −ε0/2η0, referred to as the ultra-local scales, does not contribute to the integrals in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' This yields the estimates (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9) (see equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='21) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='22) in [19] for further detail).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' It remains to show that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='11) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Applying the cumulant expansion formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='15) to the quantity E [�e(λ) {1 − E} [Gjj(z)]] yields the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7 in [17]) For all z ∈ D defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2) and j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', N} we have −1 mj(z) E [�e(λ) {1 − E} [Gjj(z)]] = − mj(z) N � k=1 Sjk E [�e(λ) {1 − E} [Gkk(z)]] − E [�e(λ) {1 − E} [Tjj(z, z)]] + E � N � k=1 SjkGkj(z)∂�e(λ) ∂Hjk � − 1 2 N � k=1 C(4) jk mj(z)mk(z) E �∂2�e(λ) ∂H2 jk � + O≺ � (1 + |λ|4) � Ψ(z)Θ(z) + N −1Ψ(z)η−1/2 0 �� , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4) where η0 is from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3), and for a, b ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=', N}, z, ζ ∈ C\\R, Txy(z, ζ) is defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Let gj := E [�e(λ) {1 − E} [Gjj(z)]] and let rj denote the right-hand side of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4) without the first term, then (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4) reads �� 1 − Sm2(z) �g � j = −mj(z)rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The operator � 1 − Sm2(z) � can be inverted to 22 deduce that gj = − �� 1 − Sm2(z) �−1 m(z)r � j, where m(z) is interpreted as a multiplication operator acting on the vector r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Summing over j, we obtain E [�e(λ) {1 − E} [Tr G(z)]] = N � j=1 gj = − N � j,k=1 �� 1 − Sm2(z) �−1� jk mk(z)r k = − N � j=1 m′ j(z) mj(z)rj, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5) where in the last step we applied the identity m′(z)/m2(z) = (1 − Sm2(z))−11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The second term on the right-hand side of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4) contributes the first term to the right hand side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='11), which, as we show in Section 6, is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Therefore, it suffices to estimate the contribution of the third and fourth terms on the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The necessary estimates on the partial derivatives of �e(λ) are collected in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6 in [17]) For all j, k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' , N} we have ∂�e(λ) ∂Hjk = −iλ π 2 1 + δjk �e(λ) � Ω′ 0 ∂ �f ∂¯ζ ∂Gkj(ζ) ∂ζ d¯ζdζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6) Moreover, for all p ∈ N, the following bound holds ���� ∂p�e(λ) ∂Hp jk ���� = O≺ � (1 + |λ|)p� , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7) and for k ̸= j ���� ∂�e(λ) ∂Hjk ���� = O≺ � N −1/2(1 + |λ|)η−1/2 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='8) Second derivatives with k ̸= j are given by ∂2�e(λ) ∂H2 jk = 2iλ π �e(λ) � Ω′ 0 ∂ �f ∂¯ζ ∂ {mj(ζ)mk(ζ)} ∂ζ d¯ζdζ + O≺ � N −1/2(1 + |λ|)2η−1/2 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9) The form in which we write the error terms in Lemmas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 slightly differs from their original form in [17] because we have already applied the estimate ∥f ′′∥1 ∼ η−1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The leading term in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9) results in the third line of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Using Lemmas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6 we proceed to estimate the third term on the right hand side of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='65) of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='8 in [17]) For all z ∈ D defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2) and all j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' , N} we have E � N � k=1 SjkGkj(z)∂�e(λ) ∂Hjk � = − 2iλ π E � �e(λ) � Ω′ 0 ∂ �f ∂¯ζ ∂Tjj(z, ζ) ∂ζ d¯ζdζ � − iλ π Sjj E [�e(λ)] � Ω′ 0 ∂ �f ∂¯ζ m′ j(ζ)mj(z)d¯ζdζ + O≺ �Ψ(z)(1 + |λ|) Nη1/2 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='10) Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' In view of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1), multiplying (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6) by SjkGkj(z), summing over k ̸= j and taking expectations gives the first term on the right hand side of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' For the remaining k = j term, observe that the function K(ζ) := Gjj(ζ) − mj(ζ) is analytic in C\\R and is stochastically dominated by Ψ(ζ) in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Applying Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5 with p = 1 to K(ζ), we obtain ∂Gjj(ζ) ∂ζ = m′ j(ζ) + O≺ � | Im ζ|−1Ψ(ζ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='11) Plugging (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='11) into (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6) with k = j and applying Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='6 with K(ζ) := ∂ζGjj(ζ) − m′ j(ζ) with s = 3/2, we get ∂�e(λ) ∂Hjj = −iλ π �e(λ) � Ω′ 0 ∂ �f ∂¯ζ m′ j(ζ)d¯ζdζ + O≺ � 1 + |λ|)N −1/2η−1/2 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='12) 23 where we used the the fact that |e(λ)| = 1 and the first line of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9) to bound |�e(λ)| by O≺(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Multiplying (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='12) by SjjGjj(z) and using the local law (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4) to estimate Gjj(z) gives the second term on the right hand side of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Application of the local law (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4) is justified by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7) with p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' This concludes the proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Summing up the leading terms in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='10) results in the second and third terms on the right-hand side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Collecting all the error terms, the estimate in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='11) now follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='13), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='5), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='9) and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' This concludes the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' References [1] Oskari Ajanki, L´aszl´o Erd˝os, and Torben Kr¨uger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Quadratic Vector Equations On Complex Upper Half-Plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Memoirs of the American Mathematical Society 261.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='1261 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' [2] Oskari Ajanki, L´aszl´o Erd˝os, and Torben Kr¨uger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Universality for general Wigner-type matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Probability Theory and Related Fields 169 (2015), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 667–727.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' [3] Oskari Ajanki, Torben Kr¨uger, and L´aszl´o Erd˝os.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Singularities of Solutions to Quadratic Vector Equations on the Complex Upper Half-Plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Communications on Pure and Applied Mathematics 70 (2017), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 1672–1705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' [4] Zhidong Bai and Jian-Feng Yao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' On the convergence of the spectral empirical process of Wigner matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Bernoulli 11 (2005), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 1059–1092.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' [5] Zhigang Bao, Kevin Schnelli, and Yuanyuan Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Central Limit Theorem for Mesoscopic Eigen- value Statistics of the Free Sum of Matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' International Mathematics Research Notices 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='7 (2020), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 5320–5382.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' [6] Zhigang Bao and Junshan Xie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' CLT for Linear Spectral Statistics of Hermitian Wigner Matrices with General Moment Conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Theory of Probability & Its Applications 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 (2016), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 187– 206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' [7] Anne Marie Boutet de Monvel and Alexei Khorunzhy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Asymptotic distribution of smoothed eigenvalue density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Gaussian random matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Random Oper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Stochastic Equations 7 (1999), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 1–22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' [8] Anne Marie Boutet de Monvel and Alexei Khorunzhy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Asymptotic distribution of smoothed eigenvalue density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Wigner random matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Random Oper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Stochastic Equations 7 (1999), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 149–168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' [9] L´aszl´o Erd˝os.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The matrix Dyson equation and its applications for random matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Random matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' IAS/Park City Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 75–158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' [10] L´aszl´o Erd˝os, Antti Knowles, and Horng-Tzer Yau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Averaging Fluctuations in Resolvents of Random Band Matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Annales Henri Poincar´e 14 (2013), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 1837–1926.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' [11] L´aszl´o Erd˝os, Antti Knowles, Horng-Tzer Yau, and Jun Yin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Delocalization and Diffusion Profile for Random Band Matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Communications in Mathematical Physics 323 (2013), 367––416.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' [12] L´aszl´o Erd˝os, Antti Knowles, Horng-Tzer Yau, and Jun Yin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The local semicircle law for a general class of random matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Electronic Journal of Probability 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='none (2013), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 1–58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' [13] Yukun He and Antti Knowles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Mesoscopic eigenvalue statistics of Wigner matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The Annals of Applied Probability 27 (2017), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 1510–1550.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' [14] Yukun He and Matteo Marcozzi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Diffusion profile for random band matrices: a short proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 177 (2019), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 666–716.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' [15] Kurt Johansson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' On fluctuations of eigenvalues of random Hermitian matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Duke Mathemat- ical Journal 91 (1998), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 151–204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' [16] Alexei Khorunzhy, Boris Khoruzhenko, and Leonid Pastur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Asymptotic properties of large ran- dom matrices with independent entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Journal of Mathematical Physics 37 (1996), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 5033– 5060.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' [17] Benjamin Landon, Patrick Lopatto, and Philippe Sosoe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Single eigenvalue fluctuations of general Wigner-type matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='01178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 24 [18] Benjamin Landon and Philippe Sosoe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Almost-optimal bulk regularity conditions in the CLT for Wigner matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='03419.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' [19] Benjamin Landon and Philippe Sosoe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Applications of mesoscopic CLTs in random matrix theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The Annals of Applied Probability 30 (2020), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 2769–2795.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' [20] Yiting Li, Kevin Schnelli, and Yuanyuan Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Central limit theorem for mesoscopic eigenvalue statistics of deformed Wigner matrices and sample covariance matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Annales de l’Institut Henri Poincar´e, Probabilit´es et Statistiques 57 (2021), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 506–546.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' [21] Yiting Li and Yuanyuan Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' On fluctuations of global and mesoscopic linear statistics of gener- alized Wigner matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Bernoulli 27 (2021), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 1057–1076.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' [22] Anna Lytova and Leonid Pastur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Central limit theorem for linear eigenvalue statistics of random matrices with independent entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' The Annals of Probability 37 (2009), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 1778–1840.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' [23] Mariya Shcherbina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Central Limit Theorem for linear eigenvalue statistics of the Wigner and sample covariance random matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Zh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Fiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 7 (2011), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 176–192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' [24] Philippe Sosoe and Percy Wong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Regularity conditions in the CLT for linear eigenvalue statistics of Wigner matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Advances in Mathematics 249 (2013), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 37–87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' [25] Eugene Wigner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Characteristics Vectors of Bordered Matrices with Infinite Dimensions II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' Annals of Mathematics 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content='2 (1957), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 203–207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'} +page_content=' 25' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAzT4oBgHgl3EQfvv6Q/content/2301.01712v1.pdf'}