Ágnes Backhausz

Ramanujan graphings and correlation decay in local algorithms

Let G be a d-regular graph of sufficiently large-girth depending on parameters k and r and µ be a random process on the vertices of G produced by a randomized local algorithm of radius r. We prove the upper bound k+1-2k/d1d-1k for the absolute value of the correlation of values on pairs of vertices of distance k and show that this bound is optimal. The same results hold automatically for factor of i.i.d processes on the d-regular tree. In that case we give an explicit description for the closure of all possible correlation sequences. Our proof is based on the fact that the Bernoulli graphing of the infinite d-regular tree has spectral radius 2d-1. Graphings with this spectral gap are infinite analogues of finite Ramanujan graphs and they are interesting on their own right.

On large‐girth regular graphs and random processes on trees

We study various classes of random processes defined on the regular tree

A random model of publication activity

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Spectral measures of factor of i.i.d. processes on vertex-transitive graphs

that are invariant under the automorphism group of

Further properties of a random graph with duplications and deletions

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Correlation Bounds for Distant Parts of Factor of IID Processes

. Most important ones are factor of i.i.d. processes (randomized local algorithms), branching Markov chains and a new class that we call typical processes. Using Glauber dynamics on processes we give a sufficient condition for a branching Markov chain to be factor of i.i.d. Typical processes are defined in a way that they create a correspondence principle between random

Local degree distributions: Examples and counterexamples

d

On the almost eigenvectors of random regular graphs

-reguar graphs and ergodic theory on

Asymptotic properties of a random graph with duplications

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Weights and Degrees in a Random Graph Model Based on 3-Interactions

. Using this correspondence principle together with entropy inequalities for typical processes we prove a family of combinatorial statements about random