Slow mixing of Glauber Dynamics for the hard-core model on regular bipartite graphs

Let $\gS=(V,E)$ be a finite, $d$-regular bipartite graph. For any $\lambda>0$ let $\pi_\lambda$ be the probability measure on the independent sets of $\gS$ in which the set $I$ is chosen with probability proportional to $\lambda^{|I|}$ ($\pi_\lambda$ is the {\em hard-core measure with activity $\lambda$ on $\gS$}). We study the Glauber dynamics, or single-site update Markov chain, whose stationary distribution is $\pi_\lambda$. We show that when $\lambda$ is large enough (as a function of $d$ and the expansion of subsets of single-parity of $V$) then the convergence to stationarity is exponentially slow in $|V(\gS)|$. In particular, if $\gS$ is the $d$-dimensional hypercube ${0,1}^d$ we show that for values of $\lambda$ tending to 0 as $d$ grows, the convergence to stationarity is exponentially slow in the volume of the cube. The proof combines a conductance argument with combinatorial enumeration methods.

💡 Research Summary

The paper investigates the mixing time of the single‑site update Markov chain (Glauber dynamics) for the hard‑core model on finite $d$‑regular bipartite graphs. For a graph $\Sigma=(V,E)$ and activity parameter $\lambda>0$, the hard‑core measure $\pi_\lambda$ assigns to each independent set $I\subseteq V$ a probability proportional to $\lambda^{|I|}$. The Glauber dynamics $M_\lambda$ proceeds by repeatedly picking a vertex uniformly at random, attempting to add it with probability $\lambda/(1+\lambda)$ or delete it with probability $1/(1+\lambda)$, and accepting the move only if the resulting set remains independent. This chain is reversible, ergodic, and has $\pi_\lambda$ as its unique stationary distribution.

The central question is how fast $M_\lambda$ converges to $\pi_\lambda$, i.e., the size of its mixing time $\tau_{M_\lambda}$. Existing results (e.g., Vigoda, Dyer‑Frieze‑Jerrum) give polynomial mixing when $\lambda$ is sufficiently small relative to the maximum degree $\Delta$, but little is known for larger $\lambda$ on regular bipartite graphs.

The authors prove that when the graph possesses a sufficiently strong bipartite expansion constant $\delta(\Sigma)$ and $\lambda$ exceeds a threshold that depends on $d$ and $\delta$, the mixing time becomes exponential in the number of vertices $|V|$. The bipartite expansion constant is defined as \