A Regularity Lemma for Bounded Degree Graphs and Its Applications: Parameter Testing and Infinite Volume Limits

We prove a regularity lemma for bounded degree graphs of subexponential growth. As an application we show that the edit-distance from a hereditary property is testable via random samplings in the graph category above. Using the regularity lemma we also prove the existence of the infinite volume limit of some log-partition functions as well as the non-randomness of the integrated density of states for random Hamiltonians in the case of aperiodic infinite graphs.

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