The Length of the Longest Increasing Subsequence of a Random Mallows Permutation

Mueller, Carl; Starr, Shannon

doi:10.1007/s10959-011-0364-5

Cited by 43 publications

(54 citation statements)

References 21 publications

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“…As explained before Theorem 1.2 above, one may again employ the idea of placing n(1 − q)/β disjoint squares of side length β/(1 − q) along the diagonal as in Figure 3. Since we expect the distribution of the points in each such square to be close to that of the Mallows µ β/(1−q),q measure, the results of [24] suggest that the typical order of magnitude of the length of the longest decreasing subsequence in each square is of order 1/ √ 1 − q. When considering decreasing subsequences we cannot concatenate the subsequences of disjoint squares, since the overall trend of the points is positive.…”

Section: Below) In Particular Lis(π) Is Distributed As Lds(π R )mentioning

confidence: 89%

“…Theorem 1.1 and the results in Section 2.1 give rigorous meaning to such statements). Thus the parameters fall in the regime of [24] and according to their results, the typical length of the longest increasing subsequence in each square is of order 1/ √ 1 − q. We may thus create an increasing subsequence with length of order n √ 1 − q by concatenating the longest increasing subsequences in each of the n(1 − q)/β squares.…”

Section: Below) In Particular Lis(π) Is Distributed As Lds(π R )mentioning

confidence: 97%

“…The proof of Theorem 1.2 proceeds along the lines of the heuristic outlined above, combining our large deviation results with the weak law of large numbers shown in [24].…”

Section: Below) In Particular Lis(π) Is Distributed As Lds(π R )mentioning

confidence: 99%

“…It is natural to allow q to be a function of n. Mueller and Starr [24] studied the regime where n(1 − q) tends to a finite limit β. They showed that LIS(π)/ √ n converges in probability to ℓ(β), where ℓ(β) is an explicitly given function of β satisfying ℓ(0) = 2 (see Theorem 5.2 for the precise statement), thus extending the results of [1,22,32].…”

Section: Below) In Particular Lis(π) Is Distributed As Lds(π R )mentioning

confidence: 99%

“…In addition, LIS(π) and LDS(π) have been studied for the Mallows distribution itself, by Mueller and Starr [24], as detailed below. We focus our investigations on the Mallows measure with q < 1.…”

Section: Introductionmentioning

confidence: 99%

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Lengths of monotone subsequences in a Mallows permutation

Bhatnagar

Peled

2014

Probab. Theory Relat. Fields

120

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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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Section: Below) In Particular Lis(π) Is Distributed As Lds(π R )mentioning

confidence: 89%

Section: Below) In Particular Lis(π) Is Distributed As Lds(π R )mentioning

confidence: 97%

“…The proof of Theorem 1.2 proceeds along the lines of the heuristic outlined above, combining our large deviation results with the weak law of large numbers shown in [24].…”

Section: Below) In Particular Lis(π) Is Distributed As Lds(π R )mentioning

confidence: 99%

Section: Below) In Particular Lis(π) Is Distributed As Lds(π R )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Lengths of monotone subsequences in a Mallows permutation

Bhatnagar

Peled

2014

Probab. Theory Relat. Fields

120

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Cycles in Mallows random permutations

Müller

Verstraaten

2023

Random Struct Algorithms

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We study cycle counts in permutations of drawn at random according to the Mallows distribution. Under this distribution, each permutation is selected with probability proportional to , where is a parameter and denotes the number of inversions of . For fixed, we study the vector where denotes the number of cycles of length in and is sampled according to the Mallows distribution. When the Mallows distribution simply samples a permutation of uniformly at random. A classical result going back to Kolchin and Goncharoff states that in this case, the vector of cycle counts tends in distribution to a vector of independent Poisson random variables, with means . Here we show that if is fixed and then there are positive constants such that each has mean and the vector of cycle counts can be suitably rescaled to tend to a joint Gaussian distribution. Our results also show that when there is a striking difference between the behavior of the even and the odd cycles. The even cycle counts still have linear means, and when properly rescaled tend to a multivariate Gaussian distribution. For the odd cycle counts on the other hand, the limiting behavior depends on the parity of when . Both and have discrete limiting distributions—they do not need to be renormalized—but the two limiting distributions are distinct for all . We describe these limiting distributions in terms of Gnedin and Olshanski's bi‐infinite extension of the Mallows model. We investigate these limiting distributions further, and study the behavior of the constants involved in the Gaussian limit laws. We for example show that as the expected number of 1‐cycles tends to —which, curiously, differs from the value corresponding to . In addition we exhibit an interesting “oscillating” behavior in the limiting probability measures for and odd versus even.

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Limit Distributions for Euclidean Random Permutations

Elboim

Peled

2019

Commun. Math. Phys.

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We study the length of cycles in the model of spatial random permutations in Euclidean space. In this model, for given length L, density ρ, dimension d and jump density ϕ, one samples ρL d particles in a d-dimensional torus of side length L, and a permutation π of the particles, with probability density proportional to the product of values of ϕ at the differences between a particle and its image under π. The distribution may be further weighted by a factor of θ to the number of cycles in π. Following Matsubara and Feynman, the emergence of macroscopic cycles in π at high density ρ has been related to the phenomenon of Bose-Einstein condensation. For each dimension d ≥ 1, we identify sub-critical, critical and super-critical regimes for ρ and find the limiting distribution of cycle lengths in these regimes. The results extend the work of Betz and Ueltschi. Our main technical tools are saddle-point and singularity analysis of suitable generating functions following the analysis by Bogachev and Zeindler of a related surrogate-spatial model.

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The Length of the Longest Increasing Subsequence of a Random Mallows Permutation

Cited by 43 publications

References 21 publications

Lengths of monotone subsequences in a Mallows permutation

Lengths of monotone subsequences in a Mallows permutation

Cycles in Mallows random permutations

Limit Distributions for Euclidean Random Permutations

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