# Phase transition for Glauber dynamics for independent sets on regular trees

R. Restrepo, D. Stefankovic, J.C. Vera Lizcano, E. Vigoda, L. Yang

Research output: Contribution to journalArticleScientificpeer-review

### Abstract

We study the effect of boundary conditions on the relaxation time (i.e., inverse spectral gap) 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 or fugacity. 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 nonreconstruction 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 trees $T_h$ of height $h$ with branching factor $b$ 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 $\Omega(n^{1+\delta/2 - o_b(1)})$. To prove this lower bound in the reconstruction region we introduce a general technique that transforms a reconstruction algorithm into a set with poor conductance.
Original language English 835-861 SIAM Journal on Discrete Mathematics 28 2 https://doi.org/10.1137/120885498 Published - 2014

### Fingerprint

Glauber Dynamics
Independent Set
Relaxation Time
Phase Transition
Hard-core Model
Boundary conditions
Branching
Local Algorithms
Sparse Graphs
Spectral Gap
Reconstruction Algorithm
Conductance
Random Graphs
Positive Solution
Markov chain
Update
Transform
Lower bound
Graph in graph theory

### Keywords

• phase transition
• mixing time
• Glauber dynamics
• Markov chain
• Monte Carlo hard-core model
• independent sets

### Cite this

Restrepo, R. ; Stefankovic, D. ; Vera Lizcano, J.C. ; Vigoda, E. ; Yang, L. / Phase transition for Glauber dynamics for independent sets on regular trees. In: SIAM Journal on Discrete Mathematics. 2014 ; Vol. 28, No. 2. pp. 835-861.
title = "Phase transition for Glauber dynamics for independent sets on regular trees",
abstract = "We study the effect of boundary conditions on the relaxation time (i.e., inverse spectral gap) 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 or fugacity. 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 nonreconstruction 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 trees $T_h$ of height $h$ with branching factor $b$ 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 $\Omega(n^{1+\delta/2 - o_b(1)})$. To prove this lower bound in the reconstruction region we introduce a general technique that transforms a reconstruction algorithm into a set with poor conductance.",
keywords = "phase transition, mixing time, Glauber dynamics, Markov chain, Monte Carlo hard-core model, independent sets",
author = "R. Restrepo and D. Stefankovic and {Vera Lizcano}, J.C. and E. Vigoda and L. Yang",
year = "2014",
doi = "10.1137/120885498",
language = "English",
volume = "28",
pages = "835--861",
journal = "SIAM Journal on Discrete Mathematics",
issn = "0895-4801",
publisher = "Society for Industrial and Applied Mathematics Publications",
number = "2",

}

Phase transition for Glauber dynamics for independent sets on regular trees. / Restrepo, R.; Stefankovic, D.; Vera Lizcano, J.C.; Vigoda, E.; Yang, L.

In: SIAM Journal on Discrete Mathematics, Vol. 28, No. 2, 2014, p. 835-861.

Research output: Contribution to journalArticleScientificpeer-review

TY - JOUR

T1 - Phase transition for Glauber dynamics for independent sets on regular trees

AU - Restrepo, R.

AU - Stefankovic, D.

AU - Vera Lizcano, J.C.

AU - Vigoda, E.

AU - Yang, L.

PY - 2014

Y1 - 2014

N2 - We study the effect of boundary conditions on the relaxation time (i.e., inverse spectral gap) 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 or fugacity. 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 nonreconstruction 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 trees $T_h$ of height $h$ with branching factor $b$ 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 $\Omega(n^{1+\delta/2 - o_b(1)})$. To prove this lower bound in the reconstruction region we introduce a general technique that transforms a reconstruction algorithm into a set with poor conductance.

AB - We study the effect of boundary conditions on the relaxation time (i.e., inverse spectral gap) 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 or fugacity. 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 nonreconstruction 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 trees $T_h$ of height $h$ with branching factor $b$ 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 $\Omega(n^{1+\delta/2 - o_b(1)})$. To prove this lower bound in the reconstruction region we introduce a general technique that transforms a reconstruction algorithm into a set with poor conductance.

KW - phase transition

KW - mixing time

KW - Glauber dynamics

KW - Markov chain

KW - Monte Carlo hard-core model

KW - independent sets

U2 - 10.1137/120885498

DO - 10.1137/120885498

M3 - Article

VL - 28

SP - 835

EP - 861

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

SN - 0895-4801

IS - 2

ER -