Logarithmic Combinatorial Structures: A Probabilistic Approach

The Mallows and Generalized Mallows models are compact yet powerful and natural ways of representing a probability distribution over the space of permutations. In this paper we deal with the problems of sampling and learning (estimating) such distributions when the metric on permutations is the Cayley distance. We propose new methods for both operations, whose performance is shown through several experiments. We also introduce novel procedures to count and randomly generate permutations at a given Cayley distance both with and without certain structural restrictions. An application in the field of biology is given to motivate the interest of this model.

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“…The Feller coupling is used to generate permutations from the uniform distribution, [2], [21] (pag. 257).…”

Section: Feller Couplingmentioning

confidence: 99%

“…We will perform a sequence of evaluations of the branch of the tree as in Figure 2 corresponding to the set of partial permutations [2 ], [24 ], [241 ]. In this process we will show how the X j are computed, the samples modified and the lower bound obtained.…”

Section: Exhaustive Algorithmmentioning

confidence: 99%

Sampling and Learning Mallows and Generalized Mallows Models Under the Cayley Distance

Irurozki

Calvo

Lozano

2016

Methodol Comput Appl Probab

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“…So in this case, there is an infinite exchangeble partition Π whose restrictions Π n are all of the Gibbs form (3). Many other combinatorially interesting examples of Gibbs partitions Π n can be given, using the prescription (3) for each fixed n: see for instance [3,24]. But typically the distributions of these combinatorially defined Π n are not consistent as n varies, so they are not realisable as the sequence of restrictions to [n] of an infinite Gibbs partition.…”

Section: Introductionmentioning

confidence: 99%

Exchangeable Gibbs partitions and Stirling triangles

2006

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For two collections of nonnegative and suitably normalised weights W = (W j ) and V = (V n,k ), a probability distribution on the set of partitions of the set {1, . . . , n} is defined by assigning to a generic partition {A j , j ≤ k} the probability V n,k W |A1| · · · W |A k | , where |A j | is the number of elements of A j . We impose constraints on the weights by assuming that the resulting random partitions Π n of [n] are consistent as n varies, meaning that they define an exchangeable partition of the set of all natural numbers. This implies that the weights W must be of a very special form depending on a single parameter α ∈ [−∞, 1]. The case α = 1 is trivial, and for each value of α = 1 the set of possible V -weights is an infinite-dimensional simplex. We identify the extreme points of the simplex by solving the boundary problem for a generalised Stirling triangle. In particular, we show that the boundary is discrete for −∞ ≤ α < 0 and continuous for 0 ≤ α < 1. For α ≤ 0 the extremes correspond to the members of the Ewens-Pitman family of random partitions indexed by (α, θ), while for 0 < α < 1 the extremes are obtained by conditioning an (α, θ)-partition on the asymptotics of the number of blocks of Π n as n tends to infinity.

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“…Associated to a Dirichlet character χ is the L-function defined for s ∈ {Re > 1} by (2). One can extend the definition to s ∈ {1/2 Re 1} by means of a functional equation (see e.g.…”

Section: Reminders On Dirichlet Characters the Content Of This Sectimentioning

confidence: 99%

Random Characters under the $L$-Measure, I: Dirichlet Characters

Barhoumi-Andréani

2017

International Mathematics Research Notices

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Abstract. We define the L-measure on the set of Dirichlet characters as an analogue of the Plancherel measure, once considered as a measure on the irreducible characters of the symmetric group.We compare the two measures and study the limit in distribution of characters evaluations when the size of the underlying group grows. These evaluations are proven to converge in law to imaginary exponentials of a Cauchy distribution in the same way as the rescaled windings of the complex Brownian motion. This contrasts with the case of the symmetric group where the renormalised characters converge in law to Gaussians after rescaling (Kerov Central Limit Theorem The goal of this article is to study the properties of characters of G q when q → +∞ and when the characters are selected at random according to the L-measure that we introduce in definition 1.1.This article focuses on the case of Dirichlet characters modulo q that are multiplicative group morphisms χ : G q → C × extended to Z/qZ by setting χ(n) = 0 if n ∈ (Z/qZ) \ (Z/qZ) × and finally extended by periodicity to Z by setting χ(n) := χ(n mod q) (see e.g. [15, § 5]). These maps were used by Dirichlet to prove his celebrated theorem on the infinitude of primes in arithmetic progressions.Dirichlet characters have the following properties :(1) χ is periodic modulo q : χ(n + q) = χ(n) for all m, n ∈ N, (2) χ is completely multiplicative :χ(n) = 0 if and only if gcd(n, q) = 1. Note that these properties imply that χ(1) = 1, as χ(1) = χ(1 × 1) = χ(1) 2 and χ(1) = 0 since gcd(1, q) = 1.We define G q to be the set of Dirichlet characters modulo q. Apart from the value 0, these characters take values in the ϕ(q)-roots of unity e 2iπk/ϕ(q) , k 0 , where ϕ is Euler's totient function. There are exactly ϕ(q) such characters.The L-function attached to a Dirichlet character χ is the following function defined for all s ∈ {Re > 1} by (see e.g. [28])We consider these functions as linear forms in the character χ, hence the choice of notation compared with the usual one that writes L(s, χ).This measure can be written in the following way using the infamous Euler formula (12) :where P is the set of prime numbers. We can thus interpret (3) in the setting of statistical mechanics as a system of particles on a circle (whose positions are given by the angles of the character evaluated in prime numbers) submitted to a particular logarithmic confinement potential at inverse temperature 2. RANDOM DIRICHLET CHARACTERS 3Let χ ≡ χ (q) denote the canonical evaluation on G q in the Dynkin formalism, namelyWe will be interested in the behaviour of the random variables χ k for a fixed integer k. The main theorem of this paper states Theorem 1.2 (Convergence in law of the evaluations). Let k ∈ 2, q be a fixed integer and s ∈ (1, +∞). Then, under P s,q , the following convergence in law is satisfiedwhere C is a standard Cauchy-distributed random variable of densityThis theorem is proven in section 3.1. An immediate striking comparison with the winding number of the complex Brownian motion can be made...

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Logarithmic Combinatorial Structures: A Probabilistic Approach

Cited by 461 publications

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Sampling and Learning Mallows and Generalized Mallows Models Under the Cayley Distance

Sampling and Learning Mallows and Generalized Mallows Models Under the Cayley Distance

Exchangeable Gibbs partitions and Stirling triangles

Random Characters under the $L$-Measure, I: Dirichlet Characters

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