1,721,051 research outputs found
Uniform Poincare inequalities for unbounded conservative spin systems: The non-interacting case
No full textAbstractWe prove a uniform Poincaré inequality for non-interacting unbounded spin systems with a conservation law, when the single-site potential is a bounded perturbation of a convex function with polynomial growth at infinity. The result is then applied to Ginzburg–Landau processes to show diffusive scaling of the associated spectral gap
Block Factorization of the Relative Entropy via Spatial Mixing
No full textWe consider spin systems in the d-dimensional lattice Zd satisfying the socalled strong spatialmixing condition. We showthat the relative entropy functional of the corresponding Gibbs measure satisfies a family of inequalities which control the entropy on a given region V subset of Z(d) in terms of a weighted sum of the entropies on blocks A subset of V when each A is given an arbitrary nonnegative weight alpha(A). These inequalities generalize thewell knownlogarithmic Sobolev inequality for the Glauber dynamics. Moreover, they provide a natural extension of the classical Shearer inequality satisfied by the Shannon entropy. Finally, they imply a family of modified logarithmic Sobolev inequalities which give quantitative control on the convergence to equilibrium of arbitrary weighted block dynamics of heat bath type
Nonlinear Dynamics for the Ising Model
No full textWe 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, Boltzmanns 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) and 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 Erdos-Renyi 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
Nonlinear Dynamics for the Ising Model
No full textWe 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(nlogn) and O(logn) 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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