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Recent Progress in Coalescent Theory
"... Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such ..."
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Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such as spin glass models. The emphasis is on recent work concerning in particular the connection of these processes to continuum random trees and spatial models such as coalescing random walks.
Invariant measures for the continual Cartan subgroup
 J. Funct. Anal
"... To Professor Malliavin with deep respect We construct and study the oneparameter semigroup of σfinite measures L θ, θ> 0, on the space of Schwartz distributions that have an infinitedimensional abelian group of linear symmetries; this group is a continual analog of the classical Cartan subgroup o ..."
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To Professor Malliavin with deep respect We construct and study the oneparameter semigroup of σfinite measures L θ, θ> 0, on the space of Schwartz distributions that have an infinitedimensional abelian group of linear symmetries; this group is a continual analog of the classical Cartan subgroup of diagonal positive matrices of the group SL(n, R). The parameter θ is the degree of homogeneity with respect to homotheties of the space, we prove uniqueness theorem for measures with given degree of homogeneity, and call the measure with degree of homogeneity equal to one the infinitedimensional Lebesgue measure L. The structure of these measures is very closely related to the socalled Poisson–Dirichlet measures PD(θ), and to the wellknown gamma process. The nontrivial properties of the Lebesgue measure are related to the superstructure of the measure PD(1), which is called the conic Poisson–Dirichlet measure – CPD. This is the most interesting σfinite measure on the set of positive convergent monotonic real series.
Cycle lengths in a permutation are typically Poisson
"... The set of cycle lengths of almost all permutations in Sn are “Poisson distributed”: we show that this remains true even when we restrict the number of cycles in the permutation. The formulas we develop allow us to also show that almost all permutations with a given number of cycles have a certain “ ..."
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The set of cycle lengths of almost all permutations in Sn are “Poisson distributed”: we show that this remains true even when we restrict the number of cycles in the permutation. The formulas we develop allow us to also show that almost all permutations with a given number of cycles have a certain “normal order” (in the spirit of the ErdősTurán theorem). Our results were inspired by analogous questions about the size of the prime divisors of “typical ” integers. 1
Invariant Measures for . . . SUBGROUP
, 2008
"... We construct and study the oneparameter semigroup of σfinite measures L θ, θ> 0, on the space of Schwartz distributions that have an infinitedimensional abelian group of linear symmetries; this group is a continual analog of the classical Cartan subgroup of diagonal positive matrices of the gro ..."
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We construct and study the oneparameter semigroup of σfinite measures L θ, θ> 0, on the space of Schwartz distributions that have an infinitedimensional abelian group of linear symmetries; this group is a continual analog of the classical Cartan subgroup of diagonal positive matrices of the group SL(n, R). The parameter θ is the degree of homogeneity with respect to homotheties of the space, we prove uniqueness theorem for measures with given degree of homogeneity, and call the measure with degree of homogeneity equal to one the infinitedimensional Lebesgue measure L. The structure of these measures is very closely related to the socalled Poisson–Dirichlet measures PD(θ), and to the wellknown gamma process. The nontrivial properties of the Lebesgue measure are related to the superstructure of the measure PD(1), which is called the conic Poisson–Dirichlet measure – CPD. This is the most interesting σfinite measure on the set of positive convergent monotonic real series.
A PROBABILISTIC INTERPRETATION OF THE MACDONALD POLYNOMIALS
"... The twoparameter Macdonald polynomials are a central object of algebraic combinatorics and representation theory. We give a Markov chain on partitions of k with eigenfunctions the coefficients of the Macdonald polynomials when expanded in the power sum polynomials. The Markov chain has stationary d ..."
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The twoparameter Macdonald polynomials are a central object of algebraic combinatorics and representation theory. We give a Markov chain on partitions of k with eigenfunctions the coefficients of the Macdonald polynomials when expanded in the power sum polynomials. The Markov chain has stationary distribution a new twoparameter family of measures on partitions, the inverse of the Macdonald weight (rescaled). The uniform distribution on cycles of permutations and the Ewens sampling formula are special cases. The Markov chain is a version of the auxiliary variables algorithm of statistical physics. Properties of the Macdonald polynomials allow a sharp analysis of the running time. In natural cases, a bounded number of steps suffice for arbitrarily large k. 1. Introduction. The Macdonald polynomials Pλ(x