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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.
Average running time of the fast Fourier transform
 J. Algorithms
, 1980
"... We compare several algorithms for computing the discrete Fourier transform of n numbers. The number of “operations ” of the original CooleyTukey algorithm is approximately 2n A(n), where A(n) is the sum of the prime divisors of n. We show that the average number of operations satisfies (l/x)Z,,,2n ..."
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We compare several algorithms for computing the discrete Fourier transform of n numbers. The number of “operations ” of the original CooleyTukey algorithm is approximately 2n A(n), where A(n) is the sum of the prime divisors of n. We show that the average number of operations satisfies (l/x)Z,,,2n A(n)(n2/9)(x2/log x). The average is not a good indication of the number of operations. For example, it is shown that for about half of the integers n less than x, the number of “operations ” is less than n i 61. A similar analysis is given for Good’s algorithm and for two algorithms that compute the discrete Fourier transform in O(n log n) operations: the chirpz transform and the mixedradix algorithm that computes the transform of a series of prime length p in O(p log p) operations. 1.
A PROBABILISTIC INTERPRETATION OF THE MACDONALD POLYNOMIALS
, 2012
"... 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.
Invariant measures for the continual Cartan 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 ..."
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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 tale of three couplings: PoissonDirichlet and GEM approximations for random permutations
 Combin. Probab. Comput
, 2006
"... Abstract. For a random permutation of n objects, as n → ∞, the process giving the proportion of elements in the longest cycle, the second longest cycle, and so on, converges in distribution to the PoissonDirichlet process with parameter 1. This was proved in 1977 by Kingman and by Vershik and Schmi ..."
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Abstract. For a random permutation of n objects, as n → ∞, the process giving the proportion of elements in the longest cycle, the second longest cycle, and so on, converges in distribution to the PoissonDirichlet process with parameter 1. This was proved in 1977 by Kingman and by Vershik and Schmidt. For soft reasons, this is equivalent to the statement that the random permutations and the PoissonDirichlet process can be coupled so that zero is the limit of the expected ℓ1 distance between the process of cycle length proporortions and the PoissonDirichlet process. We investigate how rapid this metric convergence can be, and in doing so, give two new proofs of the distributional convergence. One of the couplings we consider has an analog for the prime factorizations of a uniformly distributed random integer, and these couplings rely on the “scale invariant spacing lemma ” for the scale invariant Poisson processes, proved in this paper.
ANALYSIS OF A BOSEEINSTEIN MARKOV CHAIN
"... This paper gives sharp rates of convergence to stationarity for a Markov chain generating BoseEinstein configurations of n balls in k boxes. The analysis leads to curious identities for the arc sine distribution. 0. On a Personal Note In 1971, as a beginning graduate student at Harvard’s Departmen ..."
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This paper gives sharp rates of convergence to stationarity for a Markov chain generating BoseEinstein configurations of n balls in k boxes. The analysis leads to curious identities for the arc sine distribution. 0. On a Personal Note In 1971, as a beginning graduate student at Harvard’s Department of Statistics, I badly wanted to learn “real ” probability. Someone told me that the deepest, best book was PaulAndre Meyers ’ “Probability and Potential Theory”. For the next year and a half, three of us ran a reading group on this book. We moved slowly, like ants on a page, without any global understanding but happy to be in the presence of a master. I can’t say I internalized any abstract potential theory but I learned a lot of measure theory and the last chapter (on Choquet Theory) made a big impact on my ability to abstract deFinettis theorem. As the magisterial sequence of books by DellacherieMeyer evolved, my familiarity with the original made them welcome and accessible. I only met PaulAndre Meyer once (at Luminy in 1995). He kindly stayed around after my talk and we spoke for about an hour. I was studying rates of convergence of finite state
Cycle lengths in a permutation are typically Poisson
 Electronic Journal of Combinatorics
"... 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
doi:10.1017/S0963548305007054 Printed in the United Kingdom A Tale of Three Couplings: Poisson–Dirichlet and GEM Approximations for Random Permutations
, 2005
"... For a random permutation of n objects, as n → ∞, the process giving the proportion of elements in the longest cycle, the secondlongest cycle, and so on, converges in distribution to the Poisson–Dirichlet process with parameter 1. This was proved in 1977 by Kingman and by Vershik and Schmidt. For so ..."
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For a random permutation of n objects, as n → ∞, the process giving the proportion of elements in the longest cycle, the secondlongest cycle, and so on, converges in distribution to the Poisson–Dirichlet process with parameter 1. This was proved in 1977 by Kingman and by Vershik and Schmidt. For soft reasons, this is equivalent to the statement that the random permutations and the Poisson–Dirichlet process can be coupled so that zero is the limit of the expected 1 distance between the process of cycle length proportions and the Poisson–Dirichlet process. We investigate how rapid this metric convergence can be, and in doing so, give two new proofs of the distributional convergence. One of the couplings we consider has an analogue for the prime factorizations of a uniformly distributed random integer, and these couplings rely on the ‘scaleinvariant spacing lemma ’ for the scaleinvariant Poisson processes, proved in this paper. 1.
Factorizations
, 1997
"... Many combinatorial structures decompose into components, with the list of component sizes carrying substantial information. An integer factors into primes—this is a similar situation, but different ..."
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Many combinatorial structures decompose into components, with the list of component sizes carrying substantial information. An integer factors into primes—this is a similar situation, but different