Year: 2026
Type: preprint
Abstract: For a ferromagnetic Ising model on a graph of maximum degree $Δ\ge3$, we prove a bound of order $\sqrt n$ on every row of the influence matrix at the tree uniqueness threshold. The estimate is uniform in the degree, the external fields, and all pinnings. More generally, if the couplings are bounded by $β$ and $\varepsilon=((Δ-1)\tanhβ-1)_+$, the bound is $C(\sqrt n+n\varepsilon)$. The proof combines a pointwise cavity bound with a positive-series magnetization tilt and the field comparison theorem of Ding, Song and Sun. The critical estimate removes the logarithm in recent general graphical bounds for the ferromagnetic case. As a consequence, zero-field single-site Glauber dynamics mixes in polynomial time throughout the supercritical window $\varepsilon=O(\sqrt{\log n/n})$, with the polynomial degree depending on the window size.
Source: arXiv (Cornell University)
Author: Pei, Yan Ru
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213951872