RANDOM MATRIX MINOR PROCESSES RELATED TO PERCOLATION THEORY

The Field Theory Approach to Percolation Processes

We review the field theory approach to percolation processes. Specifically, we focus on the so-called simple and general epidemic processes that display continuous nonequilibrium active to absorbing state phase transitions whose asymptotic features are governed respectively by the directed (DP) and dynamic isotropic percolation (dIP) universality classes. We discuss the construction of a field ...

Introduction to Random-Matrix Theory

Random-matrix theory gained attention during the 1950s due to work by Eugene Wigner in mathematical physics. Specifically, Wigner wished to describe the general properties of the energy levels (or of their spacings) of highly excited states of heavy nuclei as measured in nuclear reactions (Wigner, 1957). Such a complex nuclear system is represented by an Hermitian operator H (called the Hamilto...

Determinantal point processes and random matrix theory in a nutshell

3 Universality 5 3.1 Macroscopic behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.1.1 Wigner’s semicircle law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2 Microscopic behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2.1 Bulk universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...

Random percolation as a gauge theory

Three-dimensional bond or site percolation theory on a lattice can be interpreted as a gauge theory in which the Wilson loops are viewed as counters of topological linking with random clusters. Beyond the percolation threshold large Wilson loops decay with an area law and show the universal shape effects due to flux tube quantum fluctuations like in ordinary confining gauge theories. Wilson loo...

MA3H2 Markov Processes and Percolation theory