Convergence of exclusion processes and related phenomena 
Keywords: log-Sobolev constant
algorithms
Coupling from the past
perfect sampling
simulation
Mixing times
uniform random permutations
intermediate-passage percolation
interacting particle systems
entropy
Hitting times
Markov chains
facilitated simple exclusion process

We examine convergence to equilibrium for the facilitated simple exclusion process (FSEP) on the segment with closed boundaries, as well as the symmetric FSEP on the circle. The FSEP on these spaces has degenerate rates, and exhibits transient states which the process will eventually leave to reach an ergodic component if the macroscopic density of particles is at least 1/2. If the macroscopic density is less than 1/2 the process will hit one of its absorbing states.
We first consider the symmetric FSEP on the segment [N]={1,...,N} with closed boundaries and k>N/2 particles. We show that the pre--cutoff phenomenon occurs and the mixing time has order k2 log(N-k). This agrees with the order of the mixing time for the symmetric simple exclusion process on the segment [k] with N-k particles under suitable initial conditions. We also show that the cutoff phenomenon occurs for the symmetric FSEP on the segment restricted to the ergodic component. Here the mixing time has order k2 log min(N-k,2k-N).
For the asymmetric FSEP on the segment, we show that the behaviour of the mixing time is qualitatively different from that of the asymmetric simple exclusion process on the segment; there exists initial conditions for which the hitting time of the ergodic component is exponentially slow in the number of holes N-k. In particular, when N-k is large enough, the hitting time of the ergodic component determines the mixing time. However, for the asymmetric FSEP restricted to the ergodic component, the cutoff phenomenon occurs and the mixing time has order k. This agrees with the order of the mixing time for the asymmetric simple exclusion process on the segment [k].
For the symmetric FSEP on the circle of size N, and macroscopic particle density α ∈ (1/2,1), we establish that the pre--cutoff phenomenon occurs on the ergodic component and that the order of the mixing times is N2 log N. This agrees with the order of the mixing time of the symmetric simple exclusion process on the circle of size N. We also give an upper bound on the hitting time of the ergodic component of order N2 log N for a large class of initial conditions. In particular, the mixing time of the FSEP restricted to this class of initial conditions also agrees with the order of the mixing time of the symmetric simple exclusion process on the circle of size N.
The dynamical constraint of the FSEP makes applying standard techniques for bounding the mixing time challenging. All results for the segment are obtained via comparisons to the simple exclusion process, the open boundary exclusion process and the zero--range process. We will also compare to these processes for the FSEP on the circle, although the comparisons here are less straightforward.
We then move on to consider the problem of efficiently producing a uniform random sample from the set of permutations on N objects with exactly k cycles for some 1
Our approach is based on the Feller representation of a random permutation. We design a Markov chain (specifically, an exclusion process) whose stationary distribution projects to the distribution of the Feller representation conditioned on having a given number of cycles. We show how to efficiently obtain an exact sample from this stationary distribution using an approach based on the Coupling From The Past algorithm.
We present an update scheme which allows us to update our Markov chain started from all initial conditions simultaneously, i.e.\ a grand coupling. We show that this update scheme has an interesting structure: applying the sequence of updates U1,...,Ut-1,Ut in the update scheme gives coalescence if and only if applying the reversed sequence Ut,Ut-1,...,Ut in the update scheme gives coalescence. This gives rise to an alternative form of the Coupling From The Past algorithm for a class of Markov chains.
We present a coupling for the distributions of the multiset of cycle lengths of uniform random permutations on N objects with i cycles, for all 1 ≤ i ≤ k. Samples from this coupling have the following consistency property: the sample with i+1 cycles is obtained from the sample with i cycles by breaking one of the cycles into two parts of integer size. In other words, we explicitly construct the trajectory of a fragmentation process. We then give an algorithm which obtains an exact sample from the coupling with expected run--time O(k2polylogN). This algorithm is based on an application of the Coupling From The Past algorithm to a version of the aforementioned exclusion process in which particles are assigned a priority.
Finally, we construct a directed percolation model termed intermediate--passage percolation. We show that, on the same random environment, the intermediate--passage percolation model is bounded above and below by the well--studied cases of directed first and last--passage percolation respectively. We then investigate the growth of this model and give sufficient conditions for linear growth, as a function of time on a linear scale, in all directions. The conditions we give include cases for which first or last--passage percolation do not have linear growth in all directions. We then state some open problems and give several simulations of the process.