## LEONARDO LELLI

PhD Graduate**PhD program:**: XXXVI

**supervisor**: A. Faggionato

**Thesis title:**Zero-temperature stochastic Ising model on quasi-transitive graphs

In this thesis, we examine the question of fixation for zero-temperature stochastic Ising model on some connected quasi-transitive graphs. The initial spin configuration is distributed according to a Bernoulli product measure with parameter $ p\in(0,1) $. Each vertex, at rate $ 1 $, changes its spin value if it disagrees with the majority of its neighbours and determines its spin value by a fair coin toss in case of a tie between the spins of its neighbours.
Depending on the graph where the process evolves and the initial density, the behavior of the model can be of three distinct types: if no vertex fixates the model is of type $ \mathcal I $; if all vertices fixate the model is of type $ \mathcal F $, and if there are vertices that fixate and vertices that do not, the model is called of type $ \mathcal M $. We prove that the shrink property for the underlying graph is a necessary condition in order for the zero-temperature Ising model to be of type $ \mathcal I $. This property requires that each finite set of vertices has at least one vertex whose neighborhood falls mostly outside of this set.
Our main result shows that if $ p=1/2 $ and the graph is connected, quasi-transitive, invariant under rotations and translations, then a strenghening of the shrink property, called the planar shrink property, implies that the model is of type $ \mathcal I $. Finally we prove that for one-dimensional translation invariant graphs, the shrink property is a necessary and sufficient condition for the model to be of type $ \mathcal I $.

**Research products**

11573/1692602 - 2023 -

**Zero-Temperature Stochastic Ising Model on Planar Quasi-Transitive Graphs**De Santis, E.; Lelli, L. - 01a Articolo in rivista

**paper:**JOURNAL OF STATISTICAL PHYSICS (Plenum Press:Book Customer Service, 233 Spring Street:New York, NY 10013:(212)620-8471, (212)620-8000, EMAIL: info@plenum.com, INTERNET: http://www.plenum.com, Fax: (212)807-1047) pp. - - issn: 0022-4715 - wos: (0) - scopus: 2-s2.0-85175030433 (0)