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A sequence of sparse weighted graphs G_N indexed by the number of vertices
N is said to be left-convergent if all (suitably weighted) subgraph counts
converge to a limit as N diverges. This notion generalizes in a
natural way Benjamini-Schramm's definition of local-weak convergence.
A broad research agenda aims at determining which `global' graph properties
determined by left or local-weak convergence (in other words, which
of these properties are in fact local).
We prove that, under a spectral condition on the weighted adjacency matrices
A_N of graph G_N, the free energy density of the Ising model on
this weighted graph is continuous in the left convergence topology
(hence is a local function.)
We use stochastic dynamics as a proof device.
[Based on joint work with Michael Ren]
Diffusion models are now the de facto approach to generative modeling across a wide range of data modalities including images, audio, videos, and visuomotor policies. In recent years, there has been a surge of interest in building the mathematical...