A local-degree entropy bound for systematic Gibbs scans
Year: 2026
Type: preprint
Abstract: Let \(P_1,\ldots,P_n\) be the single-site heat-bath operators of a finite probability law. Suppose that \(\operatorname{Ent}_{\mu}(g^2)\leq L\sum_{i=1}^{n}\lVert(I-P_i)g\rVert_2^2\) and that each update fails to commute with at most \(\Delta\) others. We prove that every ordered sweep contracts entropy by a factor at most \(L\Delta^2/(1+L\Delta^2)\). Approximate tensorization with constant \(C\) and a lower bound \(b>0\) on positive full one-site conditional probabilities gives \(L=C[2+2\log(1/b)]\). The estimate permits constrained support and prescribed orders that vary between sweeps. Only full one-site conditional probabilities are bounded below; no lower bound under arbitrary partial conditioning is required. The proof follows square roots of the evolving densities, bounds their terminal local residuals using commutation, and charges their squared movement to entropy loss. Applied to the established fully frustrated hypercube Ising model with \(n=2^d\) sites and fixed inverse temperature \(0<a<1/2\), the estimate gives \(O_a(d^{5/2}\log(n/\varepsilon))\) complete sweeps from every starting state, uniformly over prescribed update orders. We compare this conclusion with consequences of earlier proofs and with the faster bound available for a chosen checkerboard order. This deposit contains the preprint and its LaTeX source package, including two explanatory figures and deterministic figure-generation code. The finite computations accompanying the signed-cube figure check the illustration, not the general theorem. The manuscript includes attribution, comparisons with prior arguments, and an AI-assistance statement. Version 1.0 is prepared for journal submission; it is not a peer-reviewed or accepted publication.
Author: Vaseghi, Sam
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221451380
Cites:
Cited by:
Related to: