README: define M in Observables, drop tau_int(E) column

README.md

@@ -73,32 +73,33 @@ M = phi.mean(axis=(1, 2, 3))       # magnetisation of each configuration
 Every chain starts from a random field. One sampler iteration is one HMC trajectory
 followed by Wolff cluster flips. τ_int is the integrated autocorrelation time of the
 stored sequence in units of stored configurations, so 0.5 means consecutive
-configurations are uncorrelated. M = V⁻¹ Σ_x φ_x is the magnetisation of a
-configuration, S its action as defined in the next section, and
-E = Σ_x Σ_μ φ_x φ_{x+μ̂} its nearest-neighbour correlator.
-
-| D | L | chains × configurations | warm-up iterations | stored every | HMC acceptance | τ_int(M) | τ_int(\|M\|) | τ_int(S) | τ_int(E) |
-|---|---|---|---|---|---|---|---|---|---|
-| 2 | 16 | 1 × 10240 | 2000 | 5th iteration | 0.77 | 0.52 | 0.53 | 0.56 | 0.53 |
-| 2 | 32 | 1 × 10240 | 2000 | 5th iteration | 0.78 | 0.50 | 0.58 | 0.60 | 0.58 |
-| 2 | 64 | 1 × 10240 | 2000 | 5th iteration | 0.78 | 0.49 | 0.64 | 0.66 | 0.65 |
-| 2 | 128 | 1 × 10240 | 2000 | 5th iteration | 0.79 | 0.48 | 0.69 | 0.79 | 0.82 |
-| 3 | 16 | 1 × 2560 | 2000 | 10th iteration | 0.77 | 0.55 | 0.60 | 0.55 | 0.55 |
-| 3 | 32 | 1 × 2560 | 2000 | 10th iteration | 0.80 | 0.50 | 0.53 | 0.54 | 0.55 |
-| 3 | 64 | 1 × 2560 | 2000 | 10th iteration | 0.75 | 0.48 | 0.78 | 0.88 | 0.91 |
-| 3 | 128 | 10 × 256 | 2000 per chain | 10th iteration | 0.77 | 0.48 | 0.74 | 0.74 | 0.75 |
-
-The slowest modes of this sampler are the Z₂-even global quantities S, E, Σ_x φ_x²,
-M² and \|M\|, which relax together. In sampler iterations their τ_int grows from about
-1.3 (2D, L = 16) to 2.4 (2D, L = 128) and from 2.4 (3D, L = 16) to 4 to 7 (3D,
-L = 128), which sets the thinning above. A scan over M, \|M\|, M², S, E, Σ_x φ_x²,
-Σ_x φ_x⁴, max_x \|φ_x\|, the mode powers \|φ̃(k)\|² of the four lowest momentum shells
-and of k = (π, …, π) found no stored observable with τ_int above the larger of
-τ_int(S) and τ_int(E) by more than 0.1.
+configurations are uncorrelated. M is the magnetisation and S the action of a
+configuration, both defined in the next section.
+
+| D | L | chains × configurations | warm-up iterations | stored every | HMC acceptance | τ_int(M) | τ_int(\|M\|) | τ_int(S) |
+|---|---|---|---|---|---|---|---|---|
+| 2 | 16 | 1 × 10240 | 2000 | 5th iteration | 0.77 | 0.52 | 0.53 | 0.56 |
+| 2 | 32 | 1 × 10240 | 2000 | 5th iteration | 0.78 | 0.50 | 0.58 | 0.60 |
+| 2 | 64 | 1 × 10240 | 2000 | 5th iteration | 0.78 | 0.49 | 0.64 | 0.66 |
+| 2 | 128 | 1 × 10240 | 2000 | 5th iteration | 0.79 | 0.48 | 0.69 | 0.79 |
+| 3 | 16 | 1 × 2560 | 2000 | 10th iteration | 0.77 | 0.55 | 0.60 | 0.55 |
+| 3 | 32 | 1 × 2560 | 2000 | 10th iteration | 0.80 | 0.50 | 0.53 | 0.54 |
+| 3 | 64 | 1 × 2560 | 2000 | 10th iteration | 0.75 | 0.48 | 0.78 | 0.88 |
+| 3 | 128 | 10 × 256 | 2000 per chain | 10th iteration | 0.77 | 0.48 | 0.74 | 0.74 |
+
+The slowest modes of this sampler are the Z₂-even global quantities S, the
+nearest-neighbour correlator Σ_x Σ_μ φ_x φ_{x+μ̂}, Σ_x φ_x², M² and \|M\|, which relax
+together. In sampler iterations their τ_int grows from about 1.3 (2D, L = 16) to 2.4
+(2D, L = 128) and from 2.4 (3D, L = 16) to 4 to 7 (3D, L = 128), which sets the
+thinning above. A scan over these quantities, M, Σ_x φ_x⁴, max_x \|φ_x\|, the mode
+powers \|φ̃(k)\|² of the four lowest momentum shells and of k = (π, …, π) found no
+stored observable with τ_int above τ_int(S) by more than 0.1.
 
 ## Observables
 
-$$\chi = V\big(\langle M^2\rangle - \langle|M|\rangle^2\big), \qquad U_L = 1 - \frac{\langle M^4\rangle}{3\langle M^2\rangle^2}$$
+The magnetisation of a configuration and the quantities built from it are
+
+$$M = \frac{1}{V}\sum_x \phi_x, \qquad \chi = V\big(\langle M^2\rangle - \langle|M|\rangle^2\big), \qquad U_L = 1 - \frac{\langle M^4\rangle}{3\langle M^2\rangle^2} .$$
 
 The action S of a configuration is computed as