[Paper Review] Heat kernels are not uniform expanders

Mikołaj Frączyk, Wouter van Limbeek|arXiv (Cornell University)|May 31, 2019

Stochastic processes and statistical mechanics13 references4 citations

TL;DR

This paper proves that infinite, bounded-degree graphs cannot have expanding heat kernels—meaning the distribution of random walk positions does not exhibit uniform expansion at any scale. Using stationary random graph theory and amenability of Poisson boundaries, the authors show that such graphs must fail to be uniformly expanding, resolving a heat kernel analogue of Benjamini’s conjecture on expanders at all scales.

ABSTRACT

We study infinite analogues of expander graphs, namely graphs where subgraphs weighted by heat kernels form an expander family. Our main result is that there does not exist any infinite expander in this sense. This proves the analogue for random walks of Benjamini's conjecture that there is no infinite graph whose metric balls are uniformly expander. The proof relies on a study of stationary random graphs, in particular proving non-expansion of heat kernels in that setting. A key result is that any stationary random graph is stationary hyperfinite, which is potentially of independent interest.

Motivation & Objective

• To investigate whether infinite graphs can exhibit uniform expansion via heat kernel distributions, analogous to Benjamini’s conjecture on metric balls.
• To determine if there exists an infinite graph where the heat kernel from any root yields uniformly expanding measures over time.
• To establish that such uniform expansion via heat kernels is impossible in infinite, bounded-degree graphs.
• To develop structural results on stationary random graphs, particularly their hyperfiniteness and amenability of Poisson boundaries.
• To provide a counterexample showing that while rooted graphs can be expanding at all scales, the unrooted heat kernel cannot be uniformly expanding.

Proposed method

• Define h-expansion of heat kernels using the condition that the measure of the boundary of any set A with μⁿₒ(A) ≤ 1/2 is at least h·μⁿₒ(A).
• Use Gromov-Hausdorff convergence to show that if heat kernels are ε-expanding on a graph, they remain so on its limit graphs.
• Construct a stationary random graph as a weak-* limit of empirical measures of random walk paths on the graph.
• Prove that any stationary random graph is stationary hyperfinite, using connections to measured equivalence relations and amenability.
• Establish that the Poisson boundary of a stationary random graph is amenable, implying hyperfiniteness, which contradicts uniform expansion.
• Derive a contradiction by showing that uniform expansion would imply a sum of probabilities exceeding 1, violating stochasticity.

Experimental results

Research questions

• RQ1 Can an infinite, bounded-degree graph have heat kernels that are uniformly expanding across all scales?
• RQ2 Is there a rooted infinite graph where the heat kernel from a fixed root is uniformly expanding?
• RQ3 Do stationary random graphs admit uniformly expanding heat kernels almost surely?
• RQ4 What structural properties do stationary random graphs possess that prevent uniform expansion?
• RQ5 How does the amenability of the Poisson boundary relate to the non-expansion of heat kernels in infinite graphs?

Key findings

• There does not exist any infinite, connected, bounded-degree graph whose heat kernels are uniformly expanding.
• The heat kernel analogue of Benjamini’s conjecture on expanders at all scales is false: no such infinite graph exists.
• Any stationary random graph of bounded degree is almost surely stationary hyperfinite, a result of independent interest.
• The Poisson boundary of a stationary random graph is amenable, which implies hyperfiniteness and obstructs uniform expansion.
• The proof relies on a contradiction: assuming uniform expansion leads to a sum of probabilities exceeding 1, violating stochasticity.
• Even though rooted graphs can be expanding at all scales (e.g., via expander quotients), the unrooted heat kernel cannot be uniformly expanding.