Abstract:  In this talk, we will start by explaining a classical random reinforcement model introduced by Pólya back in 1920 which is widely known as the Pólya’s balanced urn scheme. We will then indicate several unrelated examples can be thought as following Pólya’s  model provided we allow ourselves a novel infinite dimensional generalization. After indicating the difficulties which arise due to absence of good algebraic techniques to deal with infinite dimensional matrices (or appropriate linear operators), we will explain a new probabilistic technique to handle the infinite dimensional generalization. We shall define a novel stochastic process which will turn out to be Markov with respect to a new type of “time” parameter, which instead of deterministic and unidirectional, will be random and have a tree-type structure. We shall show that this new process and the infinite dimensional generalization of Pólya’s balanced urn scheme are same in law. This will enable us to derive fairly general scaling limits for the infinite dimensional scheme and show that classical results can be easily derived without much difficulties and completely avoiding algebraic techniques. Moreover, we will show that apparently unrelated problems arising from a variety of different fields such as combinatorics, statistical physics and statistics, can be solved by using this new generalization.

[The talk will be based on several works done in the last few years. Some of which are joint work with Debleena Thacker, University of Bath, UK and Svante Jonson, Uppsala University, Sweden]

YouTube Live-Streaming Link: https://youtu.be/00JA5TwjrtE

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