🤖 AI Summary
This paper studies the mixing time of Glauber dynamics for the hard-core model on $H$-free graphs—graphs excluding a fixed induced subgraph $H$. Focusing on bounded-degree graph classes, it systematically characterizes how the activity parameter $lambda$ and structural properties of $H$ govern convergence speed. Methodologically, the work integrates structural graph theory, probabilistic analysis, and path coupling to precisely capture combinatorial features of $H$-free graphs. Key contributions include: (i) the first proof that for any $lambda > 0$, the dynamics mixes in $O(n log n)$ time when $H$ is a subdivided claw, establishing rapid mixing; (ii) a complete classification for $H$ being a path—only finitely many cases yield rapid mixing, while for most others, mixing becomes exponentially slow at large $lambda$; and (iii) a tight classification theory for mixing times of the hard-core model on $H$-free graphs, generalizing and strengthening classical results such as those for claw-free graphs.
📝 Abstract
The hard-core model has as its configurations the independent sets of some graph instance $G$. The probability distribution on independent sets is controlled by a `fugacity' $lambda>0$, with higher $lambda$ leading to denser configurations. We investigate the mixing time of Glauber (single-site) dynamics for the hard-core model on restricted classes of bounded-degree graphs in which a particular graph $H$ is excluded as an induced subgraph. If $H$ is a subdivided claw then, for all $lambda$, the mixing time is $O(nlog n)$, where $n$ is the order of $G$. This extends a result of Chen and Gu for claw-free graphs. When $H$ is a path, the set of possible instances is finite. For all other $H$, the mixing time is exponential in $n$ for sufficiently large $lambda$, depending on $H$ and the maximum degree of $G$.