MARK KAC SEMINAR

In this talk, we shall first describe the Dobrushin program of restoration of Gibbsianness in order to insist on the topological notions required to properly define Gibbs measures as equilibrium states in mathematical statistical mechanics. Our notions will be illustrated first by standard (nearest-neighbours) Ising models on the square and cubic lattices, while we shall extend our analysis to (long-range) one dimensional Ising models in order to compare Gibbs measures to a closely related family of measures introduced in dynamical systems, namely the family of g-measures. We shall describe how non-Gibbisanness helps to understand that in some phase transition region (low temperature, not too short range), the long-range Ising model provides examples of Gibbs measures which are not g-measures.

References:

1. A. Le Ny. Almost Gibbsianness and Parsimonious Description of the Decimated 2d Ising Models. JSP 152, no 2: 305–335, 2013.
2. A. van Enter and A. Le Ny. Decimation of the Dyson-Ising Ferromagnet. SPAi 127, no 11, 2017.
3. R. Bissacot, E. Endo, A. van Enter and A. Le Ny. Entropic Repulsion and Lack of the g-measure property for Dyson Models. arXiv:1705.03156

Many rare events that arise in real-life applications exhibit heavy-tailed phenomena: for example, financial losses, file sizes and delays in communication networks, and magnitudes of systemic events such as large-scale blackouts in power grids. While the theory of large deviations has been wildly successful in providing systematic tools for understanding rare events when the underlying uncertainties are light-tailed, the theory developed in the heavy-tailed setting has been mostly restricted to model-specific results or results pertaining to events that are caused by a single big jump.

In this talk, we present our recent results that go beyond such restrictions and establish sample-path large deviations for a very general class of rare events associated with heavy-tailed random walks and Levy processes. In the first part of the talk, we will introduce our main results and illustrate their implications in various applications. In particular, we will establish the principle of multiple big jumps in rigorous mathematical terms for the processes with regularly varying increments, and provide detailed answers to the open question regarding the tail asymptotics of multiple server queues with Weibull service times. In the second part of the talk, we will discuss the ideas behind the proofs of our results, a few new concepts we introduce to the general theory of M-convergence, and the connection between our results and the classical large deviation principle.