Kernekoncepter
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.
Resumé
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.