The Length of the Longest Common Subsequence of Two Independent Mallows Permutations

Jin, Ke

doi:10.48550/arxiv.1611.03840

Cited by 2 publications

(2 citation statements)

References 16 publications

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“…Theorem 3 is used in our proof (see [7]) of a weak law of large numbers for the length of the longest common subsequence of two independent Mallows permutations. Theorem 3 is also of interest in its own right because it provides us a nontrivial example for a more general question: if we have two independent sequences of random permutations {π n } and {τ n } whose limiting empirical measures are known, under what condition does the limiting empirical measure of {τ n • π n } exist with density of a similar form as ρ in (4)?…”

Section: Resultsmentioning

confidence: 99%

The Limit of the Empirical Measure of the Product of Two Independent Mallows Permutations

Jin

2019

J Theor Probab

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The Mallows measure is a probability measure on S n where the probability of a permutation π is proportional to q l(π) with q > 0 being a parameter and l(π) the number of inversions in π. We show the convergence of the random empirical measure of the product of two independent permutations drawn from the Mallows measure, when q is a function of n and n(1 − q) has limit in R as n → ∞.Keywords Mallows measure · random permutation · convergence of measure Mathematics Subject Classification (2010) 60F05 · 60B15 · 05A05 1 − q i 1 − q .

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Section: Resultsmentioning

confidence: 99%

The Limit of the Empirical Measure of the Product of Two Independent Mallows Permutations

Jin

2019

J Theor Probab

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“…It has recently enjoyed substantial interest among both pure and applied mathematicians. In particular, analysis has been carried out of the cycle structure [9] and the longest increasing subsequence [2,4,17] of a Mallows permutation, of the longest common subsequence of two independent Mallows permutations [15], and of mixing times of related Markov chains [3,6]. The Mallows permutation has also been studied as a statistical physics model [18,19], and has found applications in learning Date: 4 June 2018.…”

Section: Introductionmentioning

confidence: 99%

Mallows permutations as stable matchings

Angel,

Holroyd,

Hutchcroft

et al. 2018

Preprint

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We show that the Mallows measure on permutations of 1, . . . , n arises as the law of the unique Gale-Shapley stable matching of the random bipartite graph conditioned to be perfect, where preferences arise from a total ordering of the vertices but are restricted to the (random) edges of the graph. We extend this correspondence to infinite intervals, for which the situation is more intricate. We prove that almost surely every stable matching of the random bipartite graph obtained by performing Bernoulli percolation on the complete bipartite graph K Z,Z falls into one of two classes: a countable family (σn) n∈Z of tame stable matchings, in which the length of the longest edge crossing k is O(log |k|) as k → ±∞, and an uncountable family of wild stable matchings, in which this length is exp Ω(k) as k → +∞. The tame stable matching σn has the law of the Mallows permutation of Z (as constructed by Gnedin and Olshanski) composed with the shift k → k + n. The permutation σn+1 dominates σn pointwise, and the two permutations are related by a shift along a random strictly increasing sequence.

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The Length of the Longest Common Subsequence of Two Independent Mallows Permutations

Cited by 2 publications

References 16 publications

The Limit of the Empirical Measure of the Product of Two Independent Mallows Permutations

The Limit of the Empirical Measure of the Product of Two Independent Mallows Permutations

Mallows permutations as stable matchings

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