Duality, statistical mechanics and random matrices

Abstract

Abstract. This article will present an informal review of some results and conjectures about the spectral theory of large random matrices and related spin systems in statistical mechanics. A class of lattice spin models provides a dual representation for spectral problems in random matrix theory. Ordered and disordered phases of the spins correspond to different spectral types and quantum time evolutions. In three dimensions, we describe a phase transition for a supersymmetric statistical mechanics system inspired by random matrix theory. This transition has a classical interpretation in terms of a history dependent walk on the lattice. In the ordered phase the walk is diffusive while in the disordered phase it is localized near its starting point.

On the Manhattan pinball problem

“…This result is previously known to hold for p > 1 2 by using bond percolation, see e.g. [3], [12,Page 238]. It is suggested by physicists that the same happens for any p ∈ (0, 1) (see [3]) and the average diameter of the trajectory is conjectured to scale as exp(cp −2 ) when p tends to 0.…”

“…There are equivalent ways to state the problem formally. We follow [12,Page 238] and state it by using bond percolation. Consider the Z 2 lattice embedded into the plane R 2 .…”

Dynkin isomorphism and Mermin–Wagner theorems for hyperbolic sigma models and recurrence of the two-dimensional vertex-reinforced jump process

“…Hyperbolic sigma models were introduced as effective models to understand the Anderson transition [10,[27][28][29]32]. In Efetov's supersymmetric method [13] the expected absolute value squared of the resolvent of random band matrices, i.e., E|(H − z) −1 (i, j)| 2 where z ∈ C + and H is a random band matrix, can be expressed as a correlation function of a supersymmetric spin model.…”

“…Thus the integral of a form F is a constant multiple of the usual Lebesgue integral of the top degree part of F . A form F ∈ Ω Λ is even if the degree of all non-vanishing coefficients F I,J is even in (28). Even forms commute.…”

On the Wegner Orbital Model

“…Supersymmetric models Another perspective on the models (1.2), and random operators in general, is given by dual supersymmetric models, which were introduced by Efetov [21], following earlier work by Wegner and Schäfer [42,35]; see further the monograph of Wegner [44] and the mathematical review of Spencer [41]. In the supersymmetric approach, E|(H − z) −1 (x, y)| 2 is expressed as a two-point correlation in a dual supersymmetric model.…”