A central limit theorem for descents of a Mallows permutation and its inverse

He, Jimmy

doi:10.1214/21-aihp1167

Cited by 5 publications

(6 citation statements)

References 40 publications

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“…Aspects of the Mallows distribution that have been studied include the longest increasing subsequence [3,5,28], longest common subsequences [24], pattern avoidance [9,10,29], the number of descents [20], and the cycle structure [15].…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

“…(Almost) every even cycle must then be contained in some set

\left\{{T}_{i-1}+1,\dots, {T}_i\right\}\cup \left\{n+1-{T}_i,\dots, n-{T}_{i-1}\right\}

, and we can adapt the proof strategy that gave Theorem 1.1 to work also here. We mention that Theorem 1.2 can also be proved by starting from the center of the permutation, rather than the sides, see [21].…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

“…and we can adapt the proof strategy that gave Theorem 1.1 to work also here. We mention that Theorem 1.2 can also be proved by starting from the center of the permutation, rather than the sides, see [21].…”

Section: Sketches Of Some Ideas Used In the Proofsmentioning

confidence: 99%

“…To see the claim, note that (Π n ) [n∕2] and (Π n ) [n∕2+1,n] are independent Mallows(n∕2, q) (see e.g., Lemma 2.3 of [20]), and so can be coupled to perfectly agree with Π n∕2 and n]. However, this random variable is stochastically dominated by X 𝜏(n∕2) , since if we couple Π n with a Mallows(N, q) process, then the length of the interval is b − a = X 𝜏(n∕2) unless X 𝜏(n∕2) > n, in which case X 𝜏(n∕2) is strictly larger.…”

Section: The Proof Of Theorem 12mentioning

confidence: 99%

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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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Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

“…(Almost) every even cycle must then be contained in some set

\left\{{T}_{i-1}+1,\dots, {T}_i\right\}\cup \left\{n+1-{T}_i,\dots, n-{T}_{i-1}\right\}

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Section: Sketches Of Some Ideas Used In the Proofsmentioning

confidence: 99%

Section: The Proof Of Theorem 12mentioning

confidence: 99%

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Cycles in Mallows random permutations

Müller

Verstraaten

2023

Random Struct Algorithms

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“…A wide range of properties of the Mallows distribution has been investigated, including pattern avoidance [16,17,53], the number of descents [31], the longest monotone subsequence [7,9,49], the longest common subsequence of two Mallows permutations [38] and the cycle structure [24,32].…”

Section: Introductionmentioning

confidence: 99%

Logical limit laws for Mallows random permutations

Müller¹,

Skerman²,

Verstraaten³

2023

Preprint

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A random permutation Π n of {1, . . . , n} follows the Mallows(n, q) distribution with parameter q > 0 if P (Π n = π) is proportional to q inv(π) for all π. Here inv(π) := |{i < j : π(i) > π(j)}| denotes the number of inversions of π. We consider properties of permutations that can be expressed by the sentences of two different logical languages. Namely, the theory of one bijection (TOOB), which describes permutations via a single binary relation, and the theory of two orders (TOTO), where we describe permutations by two total orders. We say that the convergence law holds with respect to one of these languages if, for every sentence ϕ in the language, the probability P(Π n satisfies ϕ) converges to a limit as n → ∞. If moreover that limit is ∈ {0, 1} for all sentences, then the zero-one law holds.We will show that with respect to TOOB the Mallows(n, q) distribution satisfies the convergence law but not the zero-one law when q = 1, when 0 < q < 1 is fixed the zero-one law holds, and for fixed q > 1 the convergence law fails.We will prove that with respect to TOTO the Mallows(n, q) distribution satisfies the convergence law but not the zero-one law for any fixed q = 1, and that if q = q(n) satisfies 1 − 1/ log * n < q < 1 + 1/ log * n then Mallows(n, q) fails the convergence law. Here log * denotes the discrete inverse of the tower function.

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Convergence rate for the number of crossings in a random labelled tree

Santiago

Arizmendi

2023

Statistics & Probability Letters

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A central limit theorem for descents of a Mallows permutation and its inverse

Cited by 5 publications

References 40 publications

Cycles in Mallows random permutations

Cycles in Mallows random permutations

Logical limit laws for Mallows random permutations

Convergence rate for the number of crossings in a random labelled tree

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