Optimal Mixing Time of Glauber Dynamics for the Hard-Core Model on Bounded-Degree H-Free Graphs

Conceitos Básicos

The mixing time of Glauber dynamics for the hard-core model on bounded-degree graphs that exclude a fixed connected graph H as an induced subgraph exhibits a dichotomy: it is either optimal (O(n log n)) or exponential in the graph size, depending on whether H is a subdivided claw or a path.

Resumo

The content investigates the mixing time of Glauber dynamics for the hard-core model on bounded-degree graphs that exclude a fixed connected graph H as an induced subgraph.
The key findings are:

If H is a subdivided claw or a path, then the mixing time is optimal, i.e., O(n log n), where n is the number of vertices in the graph.

If H is neither a subdivided claw nor a path, then the mixing time is exponential in n for sufficiently large fugacity λ.

The analysis relies on constructing couplings that bound the Wasserstein distance between the conditional distributions of the hard-core model, which in turn implies spectral independence and optimal mixing. For graphs excluding subdivided claws, the key is to show that the clusters formed by a breadth-first search grow slowly. For graphs excluding other H, an explicit construction of graphs with exponentially small conductance is provided.
The results establish a dichotomy between optimal and exponential mixing, depending on the structure of the excluded graph H. This mirrors a similar dichotomy for the complexity of finding the largest independent set in bounded-degree H-free graphs.

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Principais Insights Extraídos De

Glauber dynamics for the hard-core model on bounded-degree $H$-free graphs

by Mark Jerrum às arxiv.org 04-12-2024

https://arxiv.org/pdf/2404.07615.pdf

Perguntas Mais Profundas

What are the implications of this dichotomy for the complexity of approximate counting and sampling problems related to the hard-core model on H-free graphs

The dichotomy established in the context has significant implications for the complexity of approximate counting and sampling problems related to the hard-core model on H-free graphs. When the excluded graph H is neither a subdivided claw nor a path, the mixing time of Glauber dynamics can vary from being optimal (O(n log n)) to exponential in the size of the graph. This dichotomy indicates that for certain classes of H-free graphs, approximate counting and sampling problems can be efficiently solved, while for others, they become computationally challenging and may require exponential time.
In practical terms, this means that for certain graph families excluding specific structures like subdivided claws or paths, algorithms for counting and sampling independent sets in the hard-core model can run efficiently with polynomial complexity. However, for graphs where these structures are not excluded, the complexity of these problems increases significantly, potentially requiring exponential time to achieve accurate results. Understanding this dichotomy is crucial for determining the computational feasibility of solving hard-core model problems on different types of graphs.

Can the techniques developed here be extended to other spin systems or models of statistical physics on H-free graphs

The techniques developed in the context can be extended to other spin systems or models of statistical physics on H-free graphs, provided that the structural properties of the excluded graph H align with the assumptions and constraints of the specific model. The key lies in analyzing the impact of the excluded graph's structure on the dynamics of the spin system and the behavior of the model on H-free graphs.
By adapting the approach of analyzing Glauber dynamics and the mixing time for the hard-core model to other spin systems, researchers can explore the complexity of approximate counting and sampling problems in various statistical physics models. This extension may involve modifying the breadth-first search procedure, investigating the conductance of the dynamics, and determining the optimal mixing time based on the specific properties of the spin system and the excluded graph H.
Overall, the techniques developed in the context can serve as a foundation for studying the computational complexity of spin systems and statistical physics models on H-free graphs, offering insights into the efficiency and feasibility of solving counting and sampling problems in these contexts.

How do the structural properties of the excluded graph H, beyond it being a subdivided claw or a path, influence the mixing time of Glauber dynamics for the hard-core model

The structural properties of the excluded graph H, beyond it being a subdivided claw or a path, play a crucial role in influencing the mixing time of Glauber dynamics for the hard-core model on H-free graphs. These properties impact the connectivity, expansion, and complexity of the graph, ultimately affecting the efficiency of the sampling and counting algorithms.
For instance, when H is a subdivided claw, the mixing time can be optimal (O(n log n)), indicating efficient convergence of the dynamics. However, for other structures excluded from the graph, such as skew stars or fork-free graphs, the mixing time may become exponential, signifying a slower convergence rate and increased computational complexity.
The specific characteristics of H, such as the presence of high-degree vertices, cycles, or unique connectivity patterns, can introduce challenges in exploring the state space of the model and transitioning between configurations. Understanding how these structural properties interact with the dynamics of the spin system is essential for predicting the mixing time and determining the computational tractability of approximate counting and sampling tasks on H-free graphs.