Related papers: Flip Dynamics for Sampling Colorings: Improving $(…

We consider Metropolis Glauber dynamics for sampling proper $q$-colourings of the $n$-vertex complete $b$-ary tree when $3\leq q\leq b/2\ln(b)$. We give both upper and lower bounds on the mixing time. For fixed $q$ and $b$, our upper bound…

We present a new lower bound on the spectral gap of the Glauber dynamics for the Gibbs distribution of a spectrally independent $q$-spin system on a graph $G = (V,E)$ with maximum degree $\Delta$. Notably, for several interesting examples,…

Let $G$ be a graph on $n$ vertices of maximum degree $\Delta$. We show that, for any $\delta > 0$, the down-up walk on independent sets of size $k \leq (1-\delta)\alpha_c(\Delta)n$ mixes in time $O_{\Delta,\delta}(k\log{n})$, thereby…

This paper studies sufficient conditions to obtain efficient distributed algorithms coloring graphs optimally (i.e.\ with the minimum number of colors) in the LOCAL model of computation. Most of the work on distributed vertex coloring so…

We consider the problem of sampling a proper $k$-coloring of a graph of maximal degree $\Delta$ uniformly at random. We describe a new Markov chain for sampling colorings, and show that it mixes rapidly on graphs of bounded treewidth if…

We establish rapid mixing of the random-cluster Glauber dynamics on random $\Delta$-regular graphs for all $q\ge 1$ and $p<p_u(q,\Delta)$, where the threshold $p_u(q,\Delta)$ corresponds to a uniqueness/non-uniqueness phase transition for…

We extend the notion of spectral independence (introduced by Anari, Liu, and Oveis Gharan [ALO20]) from the Boolean domain to general discrete domains. This property characterises distributions with limited correlations, and implies that…

This work establishes novel optimum mixing bounds for the Glauber dynamics on the Hard-core and Ising models. These bounds are expressed in terms of the local connective constant of the underlying graph $G$. This is a notion of effective…

We prove an optimal $O(n \log n)$ mixing time of the Glauber dynamics for the Ising models with edge activity $\beta \in \left(\frac{\Delta-2}{\Delta}, \frac{\Delta}{\Delta-2}\right)$. This mixing time bound holds even if the maximum degree…

Brooks' theorem states that all connected graphs but odd cycles and cliques can be colored with $\Delta$ colors, where $\Delta$ is the maximum degree of the graph. Such colorings have been shown to admit non-trivial distributed algorithms…

We analyze the mixing time of a natural local Markov chain (the Glauber dynamics) on configurations of the solid-on-solid model of statistical physics. This model has been proposed, among other things, as an idealization of the behavior of…

We consider the problem of maintaining a $(1+\epsilon)\Delta$-edge coloring in a dynamic graph $G$ with $n$ nodes and maximum degree at most $\Delta$. The state-of-the-art update time is $O_\epsilon(\text{polylog}(n))$, by Duan, He and…

Let $\epsilon \in (0, 1)$ and $n, \Delta \in \mathbb N$ be such that $\Delta = \Omega\left(\max\left\{\frac{\log n}{\epsilon},\, \left(\frac{1}{\epsilon}\log \frac{1}{\epsilon}\right)^2\right\}\right)$. Given an $n$-vertex $m$-edge simple…

For distributions over discrete product spaces $\prod_{i=1}^n \Omega_i'$, Glauber dynamics is a Markov chain that at each step, resamples a random coordinate conditioned on the other coordinates. We show that $k$-Glauber dynamics, which…

In this paper we study the classic problem of computing a maximum cardinality matching in general graphs $G = (V, E)$. The best known algorithm for this problem till date runs in $O(m \sqrt{n})$ time due to Micali and Vazirani \cite{MV80}.…

A popular method for sampling from high-dimensional distributions is the \emph{Gibbs sampler}, which iteratively resamples sites from the conditional distribution of the desired measure given the values of the other coordinates. It is…

In this work we prove sufficient conditions for the Glauber dynamics corresponding to a sequence of (non-product) measures on finite product spaces to be rapidly mixing, i.e. that the mixing time with respect to the total variation distance…

We give a new rapid mixing result for a natural random walk on the independent sets of a graph $G$. We show that when $G$ has bounded treewidth, this random walk -- known as the Glauber dynamics for the hardcore model -- mixes rapidly for…

Let $G$ be a graph of maximum degree $\Delta$ and $k$ be an integer. The $k$-recolouring graph of $G$ is the graph whose vertices are $k$-colourings of $G$ and where two $k$-colourings are adjacent if they differ at exactly one vertex. It…

The spectral independence approach of Anari et al. (2020) utilized recent results on high-dimensional expanders of Alev and Lau (2020) and established rapid mixing of the Glauber dynamics for the hard-core model defined on weighted…