Combinatorial Stochastic Processes

Combinatorial Stochastic Processes

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Jim Pitman

This is a preliminary set of lecture notes for a course to be given at the St. Flour summer school in July 2002. The theme of the course is the study of various combinatorial models of random partitions and random trees, and the asymptotics of these models related to continuous parameter stochastic processes. Following is a list of the main topics treated:

models for random combinatorial structures, such as trees, forests, permutations, mappings, and partitions;

probabilistic interpretations of various combinatorial notions e.g. Bell polynomials, Stirling numbers, polynomials of binomial type, Lagrange inversion;

Kingman's theory of exchangeable random partitions and random discrete distributions;

connections between random combinatorial structures and processes with independent increments: Poisson-Dirichlet limits;

random partitions derived from subordinators;

asymptotics of random trees, graphs and mappings related to excursions of Brownian motion;

continuum random trees embedded in Brownian motion;

Brownian local times and squares of Bessel processes;

various processes of fragmentation and coagulation, including Kingman's coalescent, the additive and multiplicative coalescents

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