Local convergence, unimodularity, and the mass-transport principle in random planar graphs
-
Vertices
-
Edges
-
Avg degree
-
Planarity
-
Mass transport balance
-
Spectral gap ≈
Graph type
Vertices N60
Root radius r2
Layout iter50
A sequence of finite graphs Gₙ converges in the Benjamini-Schramm (local/weak) sense if for every rooted graph (H,o) and radius r,
the frequency of r-neighborhoods isomorphic to (H,o) converges.
The limit is described by a unimodular random rooted graph (G,o): P(G,o) = P(G,v) for any vertex v.
Key consequence: the mass-transport principle — for any f(G,u,v)≥0:
E[Σᵥ f(G,o,v)] = E[Σᵥ f(G,v,o)].
Highlighted: ball of radius r around chosen root (purple), mass transport flows (yellow arrows).