##### Department of Mathematics,

University of California San Diego

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### Math 288 - Probability and Statistics Seminar

## Shankar Bhamidi

#### University of British Columbia

## The (Unreasonable) Effectiveness of Local Weak Convergence Methodology in Probability

##### Abstract:

The idea of local neighborhoods of probabilistic discrete structures (such as random graphs) to the local neighborhood of limiting infinite objects has been known for a long time in the probability community and has proved to be remarkably effective in proving convergence results in many different situations.\\ Here we shall give a wide range of examples of the above methodology. In particular \begin{enumerate} \item We shall show how the above methodology can be used to tackle problem of flows through random networks, where we have a random network with nodes communicating via least cost paths to other nodes. We shall show in some models how the above methodology allows us to prove the convergence of the empirical distribution of edge flows exhibiting how macroscopic order emerges from microscopic rules. \item We shall show how for a wide variety of random trees (uniform random trees, preferential attachment trees from a wide variety of attachment schemes etc), how the above methodology shows the convergence of the spectral distribution of the adjacency matrix to a limiting non random distribution function. \item Time permitting we shall also show how how one can deduce convergence of the maximal matching for various families of random trees and what this means about the spectral mass at 0. \end{enumerate} Joint work with David Aldous, Steve Evans and Arnab Sen.

### November 13, 2008

### 9:00 AM

### AP&M 6402

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