Metastability of the Ising model with random interaction coefficients

Abstract

In this Ph.D. thesis we present results on metastability of random modifications of Ising spin systems which evolve with Glauber dynamics. Their Hamiltonians have random and possibly inhomogeneous interaction coefficients. We study these models at fixed temperature, with constant external magnetic field and in the large volume limit. <br /> The stochastic modification adds a level of randomness to the models and makes the direct study of metastability usually very difficult. Therefore, our main strategy consists in studying the model with random interaction coefficients by comparison with the model where these coefficients are replaced by their expectations. Here the latter model is called annealed model. <br /> In the first part of this thesis we summarise a joint paper with A. Bovier and E. Pulvirenti (2021). Therein we studied metastability of the Ising model on the dense Erdos-Rényi random graph with constant edge probability, also called randomly dilute Curie-Weiss model (RDCW), by comparing it with the well-known Curie-Weiss model. The main novelty in the proofs is the application of Talagrand's concentration inequality to characterise the randomness of a certain generalised partition function. <br /> In the second part we prove a simple unpublished extension to more general models of the generalised partition function methods used in the first part. <br /> The third part contains a summary of a joint paper with A. Bovier and F. den Hollander (2022), in which we studied in detail metastability of an Ising model with random interaction coefficients having a product structure. The model we analysed is the annealed version of the Ising model on a Chung-Lu-like random graph with i.i.d. weights which have finite support (ICL). We provided detailed information on the metastable regime and analysed the mean metastable exit time, proving sharp asymptotic estimates and characterising its randomness up to leading order. <br /> In the last part we give an overview on how the results of the first part were extended to a wide class of spin systems with more general random interaction coefficients, in a recent joint work with A. Bovier, F. den Hollander, E. Pulvirenti and M. Slowik. This class of models includes Ising models on various inhomogeneous dense random graphs (e.g. ICL) and randomly diluted spin models. In addition to estimates on the tails of the random mean metastable exit times (showed also in the first part for RDCW), we provided estimates on their moments and conditions on metastability, always in comparison with the annealed model. The methods used include McDiarmid's inequality and novel localisation techniques developed by Schlichting and Slowik (2019).