The measurable Kesten theorem

The Measurable Kesten Theorem

We give an explicit bound on the spectral radius in terms of the densities of short cycles in finite d-regular graphs. It follows that the a finite d-regular Ramanujan graph G contains a negligible number of cycles of size less than c log log |G|. We prove that infinite d-regular Ramanujan unimodular random graphs are trees. Through Benjamini-Schramm convergence this leads to the following rigi...

The Aizenman-Kesten-Newman Theorem

The uniqueness of the infinite open cluster in the setting of bond percolation on the square grid was proven by Harris in 1960 [6]. As shown by Fisher in 1961 [4], Harris’ proof can be extended to include site percolation on the square grid. Aizenman, Kesten, and Newman [1] show that this fact is true in a much more general setting, as well. Let Λ be a connected, infinite, locally-finite, verte...

Measurable Utility and the Measurable Choice Theorem

A note on the Harris-Kesten Theorem

A short proof of the Harris-Kesten result that the critical probability for bond percolation in the planar square lattice is 1/2 was given in [1], using a sharp threshold result of Friedgut and Kalai. Here we point out that a key part of this proof may be replaced by an argument of Russo [6] from 1982, using his approximate zero-one law in place of the Friedgut-Kalai result. Russo’s paper gave ...

On the rate of convergence in the Kesten renewal theorem*

We consider the stochastic recursion Xn+1 = Mn+1Xn +Qn+1 on R , where (Mn, Qn) are i.i.d. random variables such that Qn are translations, Mn are similarities of the Euclidean space R. Under some standard assumptions the sequence Xn converges to a random variable R and the law ν of R is the unique stationary measure of the process. Moreover, the weak limit of properly dilated measure ν exists, d...