Phase Uniqueness for the Mallows Measure on Permutations

Starr, Shannon; Walters, Meg

doi:10.48550/arxiv.1502.03727

Cited by 2 publications

(4 citation statements)

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“…, with 1 ≤ a ≤ b, decreases with both b and q. The first inequality in (28) follows from the third inequality there and the fact that…”

Section: The Distribution Of the Arc Chainmentioning

confidence: 89%

“…• Lastly, if κ t = κt then κ t+1 ≤ κt+1 is a consequence of the second and third inequality in (28).…”

Section: The Distribution Of the Arc Chainmentioning

confidence: 99%

“…The Mallows distribution is a non-uniform distribution on permutations which was introduced by Mallows in statistical ranking theory [20]. It has recently been the focus of several studies in varied contexts including mixing times of Markov chains [2,9], statistical physics [27,28], learning theory [8], q-exchangeability [13,14] and the problem of the longest increasing subsequence [21,6,5]. Borodin, Diaconis and Fulman [7, Section 5] considered a class of models of random permutations (denoted P θ there) for which the Mallows distribution is the prime example.…”

Section: Introductionmentioning

confidence: 99%

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On the cycle structure of Mallows permutations

Gladkich¹,

Peled²

2018

Ann. Probab.

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We study the length of cycles of random permutations drawn from the Mallows distribution. Under this distribution, the probability of a permutation π ∈ S n is proportional to q inv(π) where q > 0 and inv(π) is the number of inversions in π.We focus on the case that q < 1 and show that the expected length of the cycle containing a given point is of order min{(1 − q) −2 , n}. This marks the existence of two asymptotic regimes: with high probability, when n tends to infinity with (1 − q) −2 ≪ n then all cycles have size o(n) whereas when n tends to infinity with (1 − q) −2 ≫ n then macroscopic cycles, of size proportional to n, emerge. In the second regime, we prove that the distribution of normalized cycle lengths follows the Poisson-Dirichlet law, as in a uniformly random permutation. The results bear formal similarity with a conjectured localization transition for random band matrices.Further results are presented for the variance of the cycle lengths, the expected diameter of cycles and the expected number of cycles. The proofs rely on the exact sampling algorithm for the Mallows distribution and make use of a special diagonal exposure process for the graph of the permutation. PermutonsThe regime of parameters in which n · (1 − q) → β is also of special interest as in this case there is a limiting density to the empirical measure of the points in the graph of a Mallows permutation. Starr [27] obtained an explicit formula for the limiting density as a function of β. In modern terminology, the limiting density is called a permuton. Recently, Mukherjee [22] proved Poisson limit theorems for the lengths of short cycles for models converging to permutons, including the Mallows model as a special case. See also Kenyon, Král', Radin and Winkler [17] for relations with permutons with fixed pattern densities.

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“…, with 1 ≤ a ≤ b, decreases with both b and q. The first inequality in (28) follows from the third inequality there and the fact that…”

Section: The Distribution Of the Arc Chainmentioning

confidence: 89%

“…• Lastly, if κ t = κt then κ t+1 ≤ κt+1 is a consequence of the second and third inequality in (28).…”

Section: The Distribution Of the Arc Chainmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the cycle structure of Mallows permutations

Gladkich¹,

Peled²

2018

Ann. Probab.

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“…One is the useful tool of insertion measures; see [16]. And finally, we have ignored work done with a single constraint because, as was noted in Section 4.3 for graphs, it does not seem to bear on the deeper issues of phases and phase transitions; however see [37] and references therein.…”

Section: Examples Of Constrained Large Permutationsmentioning

confidence: 99%

Phases in large combinatorial systems

Radin

2018

Ann. Inst. Henri Poincaré Comb. Phys. Interact.

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This is a status report on a companion subject to extremal combinatorics, obtained by replacing extremality properties with emergent structure, 'phases'. We discuss phases, and phase transitions, in large graphs and large permutations, motivating and using the asymptotic formalisms of graphons for graphs and permutons for permutations. Phase structure is shown to emerge using entropy and large deviation techniques.

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Phase Uniqueness for the Mallows Measure on Permutations

Cited by 2 publications

References 0 publications

On the cycle structure of Mallows permutations

On the cycle structure of Mallows permutations

Phases in large combinatorial systems

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