Log‐Sobolev inequality for near critical Ising models

Bauerschmidt, Roland; Dagallier, Benoit

doi:10.1002/cpa.22172

Cited by 7 publications

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Log‐Sobolev inequality for near critical Ising models

Bauerschmidt,

Dagallier

2023

Comm Pure Appl Math

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For general ferromagnetic Ising models whose coupling matrix has bounded spectral radius, we show that the log‐Sobolev constant satisfies a simple bound expressed only in terms of the susceptibility of the model. This bound implies very generally that the log‐Sobolev constant is uniform in the system size up to the critical point (including on lattices), without using any mixing conditions. Moreover, if the susceptibility satisfies the mean‐field bound as the critical point is approached, our bound implies that the log‐Sobolev constant depends polynomially on the distance to the critical point and on the volume. In particular, this applies to the Ising model on subsets of when .The proof uses a general criterion for the log‐Sobolev inequality in terms of the Polchinski (renormalisation group) equation, a recently proved remarkable correlation inequality for Ising models with general external fields, the Perron–Frobenius theorem, and the log‐Sobolev inequality for product Bernoulli measures.

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Log‐Sobolev inequality for near critical Ising models

Bauerschmidt,

Dagallier

2023

Comm Pure Appl Math

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Nonlinear Dynamics for the Ising Model

Caputo,

Sinclair

2024

Commun. Math. Phys.

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We introduce and analyze a natural class of nonlinear dynamics for spin systems such as the Ising model. This class of dynamics is based on the framework of mass action kinetics, which models the evolution of systems of entities under pairwise interactions, and captures a number of important nonlinear models from various fields, including chemical reaction networks, Boltzmann’s model of an ideal gas, recombination in population genetics and genetic algorithms. In the context of spin systems, it is a natural generalization of linear dynamics based on Markov chains, such as Glauber dynamics and block dynamics, which are by now well understood. However, the inherent nonlinearity makes the dynamics much harder to analyze, and rigorous quantitative results so far are limited to processes which converge to essentially trivial stationary distributions that are product measures. In this paper we provide the first quantitative convergence analysis for natural nonlinear dynamics in a combinatorial setting where the stationary distribution contains non-trivial correlations, namely spin systems at high temperatures. We prove that nonlinear versions of both the Glauber dynamics and the block dynamics converge to the Gibbs distribution of the Ising model (with given external fields) in times $$O(n\log n)$$ O ( n log n ) and $$O(\log n)$$ O ( log n ) respectively, where n is the size of the underlying graph (number of spins). Given the lack of general analytical methods for such nonlinear systems, our analysis is unconventional, and combines tools such as information percolation (due in the linear setting to Lubetzky and Sly), a novel coupling of the Ising model with Erdős-Rényi random graphs, and non-traditional branching processes augmented by a “fragmentation” process. Our results extend immediately to any spin system with a finite number of spins and bounded interactions.

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