Relaxation to equilibrium for a disordered particle system

A fundamental result in probability theory states that the distribution of an irreducible, time continuous Markov chain on a finite state space converges to an equilibrium as time tends to infinity. The study of Markov chain mixing time is a branch of Probability Theory that tries to understand the characteristic of this convergence and answer questions such as: How much time is needed to converge to equilibrium? Is the convergence regular or abrupt?Within this field only little effort have been made to understand how random inhomogeneities can affect the pattern of convergence to equilibrium. The objective of our research project is to develop the mathematical understanding of relaxation to equilibrium of disordered systems. To do so, we will explore in depth the case of a specific disordered model - the Simple Exclusion Process (SEP) in a random environment - with the hindsight of developing robust methods which contribute to the understanding of disordered Markov processes.Our project defines a handful of precise scientific questions on which our efforts should be focused. These concern the asymptotic behavior, as the system size go to infinity, of the spectral gap and of the mixing time for disordered Simple Exclusion Process on one dimensional graphs as well as the possible occurrence of the cutoff phenomenon for the process.