Research

Fast And Slow Mixing of Kawasaki dynamics on bounded degree graphs

Appeared in APPROX/RANDOM 2024. Here is the link to the paper.

Abstract: We study the worst-case mixing time of the global Kawasaki dynamics for the fixed-magnetization Ising model on the class of graphs of maximum degree $\Delta$. Proving a conjecture of Carlson, Davies, Kolla, and Perkins, we show that below the tree uniqueness threshold, the Kawasaki dynamics mix rapidly for all magnetizations. Disproving a conjecture of Carlson, Davies, Kolla, and Perkins, we show that the regime of fast mixing does not extend throughout the regime of tractability for this model: there is a range of parameters for which there exist efficient sampling algorithms for the fixed-magnetization Ising model on max-degree $\Delta$ graphs, but the Kawasaki dynamics can take exponential time to mix. Our techniques involve showing spectral independence in the fixed-magnetization Ising model and proving a sharp threshold for the existence of multiple metastable states in the Ising model with external field on random regular graphs.

This is joint work with Marcus Pappik, Will Perkins, and Corrine Yap.

The paper was presented at Cargese Workshop. The 30 minute version of the slides can be found here: slides.

Coloring Locally Sparse Graphs

Abstract: A graph $G$ is $k$-locally sparse if for each vertex $v \in V(G)$, the subgraph induced by its neighborhood contains at most $k$ edges. Alon, Krivelevich, and Sudakov showed that for $f > 0$ if a graph $G$ of maximum degree $\Delta$ is $\Delta^2/f$-locally-sparse, then $\chi(G) = O\left(\Delta/\log f\right)$. We introduce a more general notion of local sparsity by defining graphs $G$ to be $(k, F)$-locally-sparse for some graph $F$ if for each vertex $v \in V(G)$ the subgraph induced by the neighborhood of $v$ contains at most $k$ copies of $F$. Employing the R\"{o}dl nibble method, we prove the following generalization of the above result: for every bipartite graph $F$, if $G$ is $(k, F)$-locally-sparse, then $\chi(G) = O\left( \Delta /\log\left(\Delta k^{-1/|V(F)|}\right)\right)$. This improves upon results of Davies, Kang, Pirot, and Sereni who consider the case when $F$ is a path. Our results also recover the best known bound on $\chi(G)$ when $G$ is $K_{1, t, t}$-free for $t \geq 4$, and hold for list and correspondence coloring in the more general so-called ``color-degree'' setting.

This is joint work with James Anderson and Abhishek Dhawan.

The paper was presented in 15 minute talk at the GT Combinatorics Seminar, GSCC, and CombinaTexas and 1 hour talk at the ACO Student Seminar. Here are the slides and slides, respectively.

Here is the arxiv link.

Bicrucial k-power-free permutations

Abstract: In this work, we prove that for every $k\geq 3$ there exist arbitrarily long bicrucial $k$-power-free permutations. We also show that for every $k\geq 3$ there exist right-crucial $k$-power-free permutations of any length at least $(k-1)(2k+1)$.

This is joint work with Margarita Akhmejanova, Alexandr Valyuzhenich, and Ilya Vorobyev.

Here is the arxiv link.