For a lazy random walk on an undirected graph, the mixing time is upper bounded by the maximum hitting time. This fails for directed chains. However, I will show that for Eulerian digraphs the mixing times is $O(mn)$, where m is the number of edges and n is the number of vertices. In the reversible case, the mixing time is robust to the change of the laziness parameter. Surprisingly, in the directed setting the mixing time can be sensitive to such changes.

This is joint work with Lucas Boczkowski and Yuval Peres.

I will talk about a Markov chain on a complex of half-lines joined at a common origin, which is partially homogeneous in the sense that on each half-line a given increment distribution is used. Increment distributions are of two types: one-sided (in which the jump always moves towards, and possibly over, the origin) and symmetric. In both cases the tails are polynomial with exponent in $(0,2)$. When the walker jumps over the origin, it is routed to a new half-line according to a stochastic transition matrix. We give a criterion for recurrence or transience. This model generalizes the case of two half-lines, called the `oscillating random walk', studied by Kemperman. In the two half-line case our criterion is linear in the two tail exponents; it is only in the more general case where the non-linear nature of the criterion is revealed.

This is joint work with Dimitri Petritis (Rennes) and Mikhail Menshikov (Durham).

We introduce a natural extension of the exclusion process to hypergraphs and prove an upper bound for its mixing time. In particular we show the existence of a constant $C$ such that for any regular connected hypergraph $G$, the $\varepsilon$-mixing time of the exclusion process on $G$ with any feasible number of particles can be upper-bounded by $CT_{\text{EX}(2,G)}\log(|V|/\varepsilon)$, where $|V|$ is the number of vertices in $G$ and $T_{\text{EX}(2,G)}$ is the 1/4-mixing time of the corresponding exclusion process with 2 particles. The proof involves an adaptation of techniques invented by Morris and developed by Oliveira.

This is joint work with Stephen Connor.

We introduce a two-type internal aggregation model where the particles (oil particles and water particles) perform the following process on $\mathbb{Z}$. Starting from $n$ oils and $n$ waters at the origin, inductively if at a vertex $x$ in $\mathbb{Z}$ there are both oil and water particles, then $x$ fires one oil and one water: each particle (independently) takes one step according to simple random walk. Firing continues until at each vertex there is at most one type of particles. We establish the correct order for several statistics of the model, and identify the scaling limit under assumption of existence.

This is joint work with S. Ganguly, C. Hoffman and L. Levine.