## Next Seminar

# Probability theory for THE random graph

## Abstract

Pick two Erdös-Renyi (*n*,1/2) graphs uniformly at random. What's the chance they are the same (isomorphic)? Small. How small? Well, at most *n*!/ 2^(*n* choose 2). When *n*= 100, that's less than 10^-1300. OK, now let *n*=infinity. The chance that the two graphs are isomorphic is one (!). This is THE random graph (the Rado graph R). I will review its many non-random models and many strange properties. — It is a natural limit of the set of all finite graphs (a first order property is true for almost all finite graphs if and only if it holds with probability one in R) and this discontinuity is surprising.

In joint work with with Sourav Chatterjee we tried to find finite manifestations: For finite *n*, the largest isomorphic induced subgraph of a pair has size 4log (*n*) -2loglog(*n*)-2log(4/e) +1 (within 1, all logs base 2 in probability when n is large). This matches data amazingly well (e.g. for n more than 30) and illuminates problems in constraint satisfaction.

# A random walk on THE random graph R

## Abstract

Let *q(j)* be a probability on **N**={0,1,2,...}. Let R be a model of THE random graph. A Markov chain on **N** starts at *i* and moves to one of its neighbor *j* in R with probability proportional to *q(j)*. This Markov chain has a stationary distribution and we inquire about rates of convergence. Since each vertex is connected to half of the others and the diameter of R is 2, it seems likely that convergence is fast. In some models we show that log* (i) steps are necessary and sufficient for convergence. The proof uses a novel variant of Hardy's inequalities for trees. This is joint work with Sourav Chatterjee and Persi Diaconis.

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Zoom-link: https://us02web.zoom.us/j/81721277245?pwd=VjhadGFZcTVZamsvRkhZUExVbHAyZz09

Meeting ID 817 2127 7245

Passcode: 759491

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