The critical branching random walk is a classical model in statistical physics. It is a random subset of edges in Z^d that is obtained by drawing a critical branching process conditioned to survive forever and embedding it randomly in Z^d using random walks. It has two natural critical dimensions, 4 and 6: in dimensions at most 4 it is space filling and in dimensions at least 6 it has a well understood geometry of a random tree. Its geometry in dimension 5 remains mysterious.
In joint work with Antal Jarai we show that in dimension 5 the electrical resistance of it, viewed as an electric network, grows sublinearly in the distance (unlike dimensions 6 and above), answering a question posed by Barlow, Jarai, Kumagai and Slade. We'll discuss some applications of this to the study of random walks on it and present some open problems.
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