Abstract
On the space of spin configurations on the 3d square lattice, we consider the shaken dynamics, a parallel Markovian dynamics that can be interpreted in terms of Probabilistic Cellular Automata. The transition probabilities are defined in terms of pair ferromagnetic Ising-type Hamiltonians with nearest neighbor interaction J, depending on an additional parameter q, measuring the tendency of the system to remain locally in the same state. Odd times and even times have different transition probabilities. We compute the stationary measure of the shaken dynamics and we investigate its relation with the Gibbs measure for the 3d Ising model. It turns out that the two parameters J and q tune the geometry of the underlying lattice. We conjecture the existence of unique line of critical points in plane. By a judicious use of perturbative methods we delimit the region where such curve must lie and we perform numerical simulation to determine it. Our method allows us to find in a unified way the critical values of J for Ising model with first neighbors interaction, defined on a whole class of lattices, intermediate between the two-dimensional hexagonal and the three-dimensional cubic one, such as, for example, the tetrahedral lattice. Finally we estimate the critical exponents of the magnetic susceptibility and show that our model captures a dimensional transition in the geometry of the system at .
Funding Statement
BS acknowledges the support of the Italian MIUR Department of Excellence grant (CUP E83C18000100006). AT acknowledges the support of the H2020 Project Stable and Chaotic Motions in the Planetary Problem (Grant 677793 StableChaoticPlanetM of the European Research Council).
Citation
Benedetto Scoppola. Alessio Troiani. Matteo Veglianti. "Shaken dynamics on the 3d cubic lattice." Electron. J. Probab. 27 1 - 26, 2022. https://doi.org/10.1214/22-EJP803
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