Phase Transition for Glauber Dynamics for Independent Sets on Regular Trees

We study the effect of boundary conditions on the relaxation time of the Glauber dynamics for the hard-core model on the tree. The hard-core model is defined on the set of independent sets weighted by a parameter $\lambda$, called the activity. The Glauber dynamics is the Markov chain that updates a randomly chosen vertex in each step. On the infinite tree with branching factor $b$, the hard-core model can be equivalently defined as a broadcasting process with a parameter $\omega$ which is the positive solution to $\lambda=\omega(1+\omega)^b$, and vertices are occupied with probability $\omega/(1+\omega)$ when their parent is unoccupied. This broadcasting process undergoes a phase transition between the so-called reconstruction and non-reconstruction regions at $\omega_r\approx \ln{b}/b$. Reconstruction has been of considerable interest recently since it appears to be intimately connected to the efficiency of local algorithms on locally tree-like graphs, such as sparse random graphs. In this paper we show that the relaxation time of the Glauber dynamics on regular $b$-ary trees $T_h$ of height $h$ and $n$ vertices, undergoes a phase transition around the reconstruction threshold. In particular, we construct a boundary condition for which the relaxation time slows down at the reconstruction threshold. More precisely, for any $\omega \le \ln{b}/b$, for $T_h$ with any boundary condition, the relaxation time is $\Omega(n)$ and $O(n^{1+o_b(1)})$. In contrast, above the reconstruction threshold we show that for every $\delta>0$, for $\omega=(1+\delta)\ln{b}/b$, the relaxation time on $T_h$ with any boundary condition is $O(n^{1+\delta + o_b(1)})$, and we construct a boundary condition where the relaxation time is $\Omega(n^{1+\delta/2 - o_b(1)})$.

💡 Research Summary

The paper investigates how boundary conditions affect the mixing time of the Glauber dynamics for the hard‑core model on regular (b)-ary trees. The hard‑core model assigns a weight (\lambda^{|I|}) to each independent set (I) and can be equivalently described by a broadcasting process with parameter (\omega) satisfying (\lambda=\omega(1+\omega)^b). In this process a child is occupied with probability (\omega/(1+\omega)) when its parent is unoccupied, and is forced to be empty otherwise. The broadcasting process exhibits a reconstruction phase transition at (\omega_r\approx\ln b/b): for (\omega>\omega_r) the state of the root can be inferred from the configuration on a distant boundary (reconstruction), while for (\omega\le\omega_r) the influence of the root decays exponentially (non‑reconstruction).

The authors prove that this information‑theoretic threshold coincides with a sharp change in the spectral gap of the Glauber dynamics, i.e., in its relaxation time. Their results can be summarized as follows.

1. Non‑reconstruction regime ((\omega\le\omega_r)).
   For any boundary condition on a tree of height (h) with (n) vertices, the relaxation time is at least (\Omega(n)) and at most (O!\bigl(n^{1+o_b(1)}\bigr)). The (o_b(1)) term tends to zero as the branching factor (b) grows. In this regime the correlation between the root and the leaves decays exponentially fast, so a single‑site update propagates information throughout the tree in essentially linear time. Consequently, the Glauber chain mixes quickly regardless of how the leaves are fixed.

2. Reconstruction regime ((\omega>\omega_r)).
   Fix a constant (\delta>0) and set (\omega=(1+\delta)\ln b/b). For every boundary condition the relaxation time is bounded above by (O!\bigl(n^{1+\delta+o_b(1)}\bigr)). However, the authors construct a specific “worst‑case” boundary—alternating blocks of all‑occupied and all‑unoccupied leaves that maximize the dependence between the root and the boundary—under which the relaxation time is bounded below by (\Omega!\bigl(n^{1+\delta/2-o_b(1)}\bigr)). Thus, above the reconstruction threshold the mixing time can become super‑linear, and the exact exponent depends on the choice of boundary.

The technical core of the paper combines several probabilistic and spectral tools. First, a conductance (Cheeger) argument yields lower bounds on the spectral gap by exhibiting a bottleneck set of configurations that is hard to leave when the boundary induces strong root‑leaf correlation. Second, an upper bound is obtained via a comparison with a simple random walk on the state space, using canonical paths whose length is proportional to the tree height. The analysis of these paths relies on a recursive description of marginal occupation probabilities on each level of the tree. Third, the authors exploit the known correlation‑decay properties of the broadcasting process: in the non‑reconstruction regime the influence of the root on a leaf decays as ((\omega b)^k) with distance (k), while in the reconstruction regime it stays bounded away from zero, creating a persistent “signal” that slows down the dynamics.

These results provide a concrete, rigorous confirmation of the conjectured equivalence between the reconstruction threshold and the algorithmic threshold for local Markov chains on locally tree‑like graphs. Since sparse random graphs (e.g., (G(n,d/n))) locally converge to a Galton–Watson tree with branching factor (d), the findings imply that Glauber dynamics for the hard‑core model on such graphs will also experience a dramatic slowdown once the activity (\lambda) crosses the corresponding critical value. This has direct implications for the design of efficient sampling algorithms in statistical physics, combinatorial optimization, and probabilistic inference.

Beyond the hard‑core model, the methodology suggests a pathway to study similar phase transitions for other spin systems (Ising, Potts, coloring) where a broadcasting representation exists. Future work could explore time‑varying or random boundary conditions, extensions to non‑regular trees, and the impact of external fields. Overall, the paper makes a significant contribution by linking a deep information‑theoretic phenomenon (reconstruction) to a concrete computational metric (mixing time) through precise spectral analysis on trees.