arXiv subjects

Victor Dubach

Publications and source records attributed to Victor Dubach.

6 recordsLinked to original sources

arXiv2026

Large deviation principles for pattern-avoiding permutations, and limit shapes for constrained Mallows permutations

We study Mallows random permutations conditioned to avoid a given pattern $\alpha$ of length~$3$. When the bias parameter is of the form $e^{\beta/n}$, we prove that these permutations converge to a non-trivial explicit deterministic permuton that depends on the pattern $\alpha$ and on the parameter $\beta$. Along the way, we provide parametrizations for $\alpha$-avoiding permutons, and establish a large deviation principle for uniform $\alpha$-avoiding permutations. As a byproduct of the proof, we also obtain asymptotic estimates of two versions of $q$-Catalan numbers in the regime $q=e^{\beta/n}$.

math.PR↗

arXiv2025

Binary search trees of permuton samples

Binary search trees (BST) are a popular type of data structure when dealing with ordered data. Indeed, they enable one to access and modify data efficiently, with their height corresponding to the worst retrieval time. From a probabilistic point of view, binary search trees associated with data arriving in a uniform random order are well understood, but less is known when the input is a non-uniform random permutation. We consider here the case where the input comes from i.i.d. random points in the plane with law $μ$, a model which we refer to as a permuton sample. Our results show that the asymptotic proportion of nodes in each subtree depends on the behavior of the measure $μ$ at its left boundary, while the height of the BST has a universal asymptotic behavior for a large family of measures $μ$. Our approach involves a mix of combinatorial and probabilistic tools, namely combinatorial properties of binary search trees, coupling arguments, and deviation estimates.

math.PR↗

arXiv2024

Locally uniform random permutations with large increasing subsequences

We investigate the maximal size of an increasing subset among points randomly sampled from certain probability densities. Kerov and Vershik's celebrated result states that the largest increasing subset among $N$ uniformly random points on $[0,1]^2$ has size asymptotically $2\sqrt{N}$. More generally, the order $Θ(\sqrt{N})$ still holds if the sampling density is continuous. In this paper we exhibit two sufficient conditions on the density to obtain a growth rate equivalent to any given power of $N$ greater than $\sqrt{N}$, up to logarithmic factors. Our proofs use methods of slicing the unit square into appropriate grids, and investigating sampled points appearing in each box.

math.PR↗

arXiv2024

Classical patterns in Mallows permutations

We study classical pattern counts in Mallows random permutations with parameters $(n,q_n)$, as $n\to\infty$. We focus on three different regimes for the parameter $q = q_n$. When $n^{3/2}(1-q)\to0$, we use coupling techniques to prove that pattern counts in Mallows random permutations satisfy a central limit theorem with the same asymptotic mean and variance as in uniformly random permutations. When $q\to1$ and $n(1-q)\to\infty$, we use results on the displacements of permutation points to find the order of magnitude of pattern counts. When $q\in(0,1)$ is fixed, we use the regenerative property of the Mallows distribution to compare pattern counts with certain $U$-statistics, and establish central limit theorems. We also construct a specific Mallows process, that is a coupling of Mallows distributions with $q$ ranging from $0$ to $1$, for which the process of pattern counts satisfies a functional central limit theorem.

math.PR↗

arXiv2024

Increasing subsequences of linear size in random permutations and the Robinson-Schensted tableaux of permutons

The study of longest increasing subsequences (LIS) in permutations led to that of Young diagrams via Robinson-Schensted's (RS) correspondence. In a celebrated paper, Vershik and Kerov obtained a limit theorem for such diagrams and found that the LIS of a uniform permutation of size n behaves as $2\sqrt{n}$. Independently and much later, Hoppen et al. introduced the theory of permutons as a scaling limit of permutations. In this paper, we extend in some sense the RS correspondence of permutations to the space of permutons. When the "RS-tableaux" of a permuton are non-trivial, we show that the RS-tableaux of random permutations sampled from this permuton exhibit a linear behavior, in the sense that their first rows and columns have lengths of linear order. In particular, the LIS of such permutations behaves as a multiple of n. We also prove some large deviation results for these convergences. Finally, by studying asymptotic properties of Fomin's algorithm for permutations, we show that the RS-tableaux of a permuton satisfy a partial differential equation.

math.PR↗

arXiv2024

A geometric approach to conjugation-invariant random permutations

We propose a new approach to conjugation-invariant random permutations. Namely, we explain how to construct uniform permutations in given conjugacy classes from certain point processes in the plane. This enables the use of geometric tools to study various statistics of such permutations. For their longest decreasing subsequences, we prove universality of the $2\sqrt n$ asymptotic. For Robinson--Schensted shapes, we prove universality of the Vershik--Kerov--Logan--Shepp limit curve, thus solving a conjecture of Kammoun. For the number of records, we establish a phase transition phenomenon as the number of fixed points grows. For pattern counts, we obtain an asymptotic normality result, partially answering a conjecture of Hamaker and Rhoades.

math.PR↗