Dan Romik (Weizmann I.)

will speak on

Limit shapes of random partitions.

Abstract:

A partition of n is a representation n = a_1 + a_2 + ... a_k of n as a sum of integers arranged in decreasing order. A partition may be drawn in a natural way as the graph of a decreasing function on [0,oo), the so-called Young diagram. A theorem of A. Vershik says that for a "typical" partition of a large integer n (that is, for a fraction of partitions of n that tends to 1 as n-->oo), the Young diagram will resemble a scaling of the function f(x) = -log(1-exp(-cx))/c, where c=pi/sqrt(6), in other words random partitions have asymptotically a limit shape given by the function f(x). I will show how this result is derived and how it can be extended to treat various sub-classes of partitions defined by imposing restrictions, such as: partitions without repeated numbers, partitions without two consecutive numbers etc. so each such sub-class there is a formula for the limit shape that describes the behavior of a "typical" element in the sub-class.

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Last update on 20/10/02.