## Shaken dynamics: metastable behavior and applications

The Shaken Dynamics ([3]) is a Markovian parallel dynamics for spin systems on arbitrary graphs with nearest neighbors interaction described by a Hamiltonian function H(σ). The transition probabilities of this dynamics are defined in terms of a pair Hamiltonian H(σ,τ) = \sum_x h_x(σ)τ_x where h_x(σ) depends on the value of the spins in a neighborhood of x and on the value of the spin at site x itself. Each transition of the Shaken Dynamics is obtained by combining two irreversible elementary steps. Despite the irreversible elementary steps, the dynamics turns out to be reversible. In this talk, I will describe the stationary measure of the Shaken Dynamics and show how it relates to the Gibbs Measure ([2]). Further, in the case of Z^2, I will compare the tunneling times from the metastable to the stable state between the Shaken Dynamics, a symmetric PCA, and a single spin-flip dynamics. Finally, I will show how the Shaken dynamics can be used to define a natively parallel algorithm to face problems in combinatorial optimization ([1]). Results are based on joint works with V. Apollonio, R. D’Autilia, B. Scoppola and E. Scoppola. References [1] B. Scoppola, A. Troiani, Gaussian Mean Field Lattice Gas, J Stat Phys, 170(2018), 1161–1176. [2] V. Apollonio, R. D’Autilia, B. Scoppola, E. Scoppola, A. Troiani, Criticality of Measures on 2-d Ising Configurations: From Square to Hexagonal Graphs, J Stat Phys 177, 1009-1021 (2019) [3] V. Apollonio, R. D’Autilia, B. Scoppola, E. Scoppola, A. Troiani, Shaken Dynamics: An Easy Way to Parallel Markov Chain Monte Carlo, J Stat Phys 189, 39 (2022)

Area: IS5 - Metastability (Elena Pulvirenti)

Keywords: Probabilistic Cellular Automata, Markov Chain Monte Carlo, Combinatorial Optimization, QUBO,

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