Colloquium WS 2025/26

Speaker: Giambattista Giacomin, Università degli Studi di Padova – Invited by H. Schulz-Baldes

Abstract: The transfer matrix method is a standard method to solve (in particular) one dimensional statistical mechanics models. In the case of spins taking only two values (Ising spins) the matrix involved is just a two by two matrix. In a somewhat surprising way and, for some cases, in highly non trivial way, the Ising transfer matrix is key to the solution of other models, notably dimer models and two dimensional Ising models. While for non disordered models everything boils down to computing the spectrum of the transfer matrix, in presence of disorder one needs to understand the much less explicit Lyapunov spectrum of the product of random Ising transfer matrices. In this field a number of mathematical challenges emerge from some remarkable research works in theoretical physics that, in mathematical terms, address the problem of singular behavior of Lyapunov exponents. The purpose of my presentation is to give an introduction to this research domain and present some rigorous results addressing precisely some of the issues tackled by physicists.