Permutations with fixed pattern densities

Kenyon, Richard; Kráľ, Daniel; Radin, Charles; Winkler, Peter

doi:10.1002/rsa.20882

Cited by 33 publications

(45 citation statements)

References 36 publications

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“…Proof of Proposition 3.3 and Lemma 3.4. The newly revealed portions of Γ π at time t+1 were described in (17). Denote the first two of these portions by…”

Section: Diagonal Exposure and The Arc Chain Processmentioning

confidence: 99%

“…Let us describe how A t evolves during the diagonal exposure process, i.e., the relationship between A t and A t+1 ; see Figure 5 for an example. The newly exposed portions of the graph Γ π at time t + 1 were described in (17). Thus • If κ t+1 = κ t then either t + 1 is a fixed point of π or t + 1 extends an open arc in A t to a longer open arc in A t+1 , either as the head or as the tail of the arc.…”

Section: Main Theoremsmentioning

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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“…Proof of Proposition 3.3 and Lemma 3.4. The newly revealed portions of Γ π at time t+1 were described in (17). Denote the first two of these portions by…”

Section: Diagonal Exposure and The Arc Chain Processmentioning

confidence: 99%

Section: Main Theoremsmentioning

confidence: 99%

On the cycle structure of Mallows permutations

Gladkich¹,

Peled²

2018

Ann. Probab.

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“…Permutons are then defined more generally to be probability measures on [0, 1] 2 with uniform marginals. We put the usual weak topology of measure theory (or weak * topology from functional analysis) on the compact space M of permutons, and note that the set ∪ n {γ π | π ∈ S n } is dense in M [13,14,12,16].…”

Section: Phases In Large Constrained Permutations 51 Permutons and Ementioning

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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“…The limiting distribution of the points of a random permutation is known as a permuton (cf. Hoppen et al (2013)) and has recently been studied in the context of finding the limiting distribution of permutation statistics such as cycle lengths Mukherjee et al (2016) and certain limit shapes of permutations with fixed pattern densities Kenyon et al (2015).…”

Section: Introductionmentioning

confidence: 99%

The length of the longest common subsequence of two independent mallows permutations

Jin¹

2019

Ann. Appl. Probab.

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The Mallows measure is a probability measure on Sn 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 prove a weak law of large numbers for the length of the longest common subsequences 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 → ∞.

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Permutations with fixed pattern densities

Cited by 33 publications

References 36 publications

On the cycle structure of Mallows permutations

On the cycle structure of Mallows permutations

Phases in large combinatorial systems

The length of the longest common subsequence of two independent mallows permutations

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