Arvind Ayyer

We study an $n$-species $t$-PushTASEP, an integrable long-range stochastic process, on a one-dimensional periodic lattice with inhomogeneities $x_1,\ldots,x_L$ and arbitrary capacity $l$ at each lattice site. The Markov matrix is identified with an alternating sum of commuting transfer matrices over all fundamental representations of $U_t(\widehat{sl}_{n+1})$. Stationary probabilities are expressed in a matrix product form involving a fusion of quantized corner transfer matrices for the strange five-vertex model introduced by Okado, Scrimshaw, and the second author. The resulting partition function, which serves as the normalization factor of the stationary probabilities, is obtained from the $l=1$ case by a finite plethystic substitution of length $l$.

The misanthrope process is an interacting particle system where particles move between neighbouring sites with hop rates depending only on the number of particles at the departure and arrival sites. Motivated by a discretised version of the Hammersley--Aldous--Diaconis process, we introduce a partially asymmetric long range misanthrope process (PALRMP) on a finite one-dimensional lattice with periodic boundary conditions where particles can move between sites that are not necessarily neighbours, as long as there are no particles in between the departure and arrival sites. In this model, each site $\ell$ has an inhomogeneous rate parameter $x_\ell$ associated to it, and the hop rate of a particle moving from site $k$ to site $\ell$ depends upon the parameter associated to the target site $x_\ell$, the direction the particle moves, and the number of particles at sites $k$ and $\ell$. We also consider the homogeneous PALRMP, where all the $x_\ell$'s are 1. We find necessary and sufficient conditions on the hop rates under which the stationary distribution is of factorised form for both the PALRMP and the homogeneous PALRMP, as well as the extreme variants, namely the ones where the particle motion is totally asymmetric (TALRMP) and symmetric (SLRMP). As an illustrative example, we study in detail the discrete Hammersley--Aldous--Diaconis process.

The well known bunkbed conjecture about percolation on finite graphs is now resolved; Gladkov, Pak and Zimin, building upon work of Hollom, have constructed a counterexample. We revisit this conjecture and study it in the broader context of the class of random cluster measures. We show that the major partial (positive) results on the bunkbed conjecture can also be proved for all random cluster measures, including the results for complete graphs, complete bipartite graphs, and the case when $p \uparrow 1$. The arboreal gas measure for forests is another limit of the random cluster measure for which we conjecture the inequality to be true and provide proofs in special cases. We identify a setting where the conjecture does hold, that of ``almost spanning tree measures''. A further analysis leads to intriguing correlation inequalities that complement Rayleigh's inequalities for spanning tree measures.