Thermodynamic limit for the Mallows model on Sn

We study the length of the longest increasing and longest decreasing subsequences of random permutations drawn from the Mallows measure. Under this measure, the probability of a permutation π ∈ S n is proportional to q Inv(π) where q is a real parameter and Inv(π) is the number of inversions in π. The case q = 1 corresponds to uniformly random permutations. The Mallows measure was introduced by Mallows in connection with ranking problems in statistics.We determine the typical order of magnitude of the lengths of the longest increasing and decreasing subsequences, as well as large deviation bounds for them. We also provide a simple bound on the variance of these lengths, and prove a law of large numbers for the length of the longest increasing subsequence. Assuming without loss of generality that q < 1, our results apply when q is a function of n satisfying n(1 − q) → ∞. The case that n(1 − q) = O(1) was considered previously by Mueller and Starr. In our parameter range, the typical length of the longest increasing subsequence is of order n √ 1 − q, whereas the typical length of the longest decreasing subsequence has four possible behaviors according to the precise dependence of n and q.We show also that in the graphical representation of a Mallows-distributed permutation, most points are found in a symmetric strip around the diagonal whose width is of order 1/(1 − q). This suggests a connection between the longest increasing subsequence in the Mallows model and the model of last passage percolation in a strip.

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“…What is the form of this density? Starr [29] has answered this question in the regime where n(1 − q) tends to a finite constant.…”

Section: (Rsk Correspondence)mentioning

confidence: 99%

“…In this regime Starr [29] developed a Botzmann-Gibbs formulation of the Mallows measure and found a limiting density for the graphical representation of the random permutation. Mueller and Starr relied on this limiting density and applied similar techniques to those of Deuschel and Zeitouni [10] to find the leading behavior of E(LIS(π)).…”

Section: Techniquesmentioning

confidence: 99%

Lengths of monotone subsequences in a Mallows permutation

Bhatnagar

Peled

2014

Probab. Theory Relat. Fields

120

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“…The literature on Mallows and Generalized Mallows models ranges from theoretical discussion [15], [17], [19], [24], [42], [43] to more practical applications to multilabel classification [9], label ranking [8] or estimation of distribution algorithms [7].…”

Section: Introductionmentioning

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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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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“…Mallows' distributions under which π n is conditionally uniform given the number of inversions were studied in [12,30]. In [29] stat is the major index of permutation.…”

Section: Introductionmentioning

confidence: 99%

Coherent random permutations with biased record statistics

Gnedin

2011

Discrete Mathematics

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a b s t r a c tWe consider random permutations that are defined coherently for all values of n, and for each n have a probability distribution which is conditionally uniform given the set of upper and lower record values. Our central example is a two-parameter family of random permutations that are conditionally uniform given the counts of upper and lower records. This family may be seen as an interpolation between two versions of Ewens' distribution. We discuss characterisations of the conditionally uniform permutations, their asymptotic properties, constructions and relations to random compositions.

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Thermodynamic limit for the Mallows model on Sn

Cited by 58 publications

References 17 publications

Lengths of monotone subsequences in a Mallows permutation

Lengths of monotone subsequences in a Mallows permutation

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

Coherent random permutations with biased record statistics

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