A disorder-independent tanh law for parity statistics in binary linear codes under Glauber dynamics

Abstract

We consider binary linear codes defined by a full-rank parity-check matrix H ∈ Fm×N 2 whose row space contains the all-ones vector, endowed with the Glauber (single-bit-flip Metropolis) dynamics at inverse temperature y on the energy E(x) = |Hx|, the Hamming weight of the syndrome. We prove that the total parity coordinate of the syndrome thermalizes as an isolated two-level system with unit gap: Podd(y) = (1 − tanh(y/2))/2 = 1/(1 + ey ), independently of every other detail of H. We also prove an exact shell identity: among the (m w) syndromes of weight w, the fraction with odd total parity is exactly w/m. Both results are verified by exhaustive enumeration of all 213 syndromes of a concrete 13 × 137 code, agreeing with the closed forms to absolute deviations below 10−78 in 80-digit arithmetic. The law provides an exact, self-calibrating thermometer for simulated-annealing decoders and a rare example of an exact annealed identity outside the Nishimori line.

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