Shaken dynamics: metastable behavior and applications

Apollonio Valentina, Università di Roma "Tre"
D'Autilia Roberto, Università di Roma "Tre"
Scoppola Elisabetta, Università di Roma "Tre"
Scoppola Benedetto, University of Rome Tor Vergata
Troiani Alessio, Università degli Studi di Perugia

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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