Talks

Long increasing subsequences in Brownian-type permutations

What is the behavior of the longest increasing subsequence of a uniformly random permutation? Its length is of order  plus Tracy–Widom fluctuations of order . Its scaling limit is the directed geodesic of the directed landscape.  This talk discusses how this behavior changes dramatically when one looks at universal Brownian-type permutations, i.e., permutations sampled from the Brownian separable permutons. We show that there are explicit constants such that …

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Meanders and Meandric Systems

In 1912 Henri Poincaré asked the following simple question: “In how many different ways a simple loop in the plane, called a meander, can cross a line a specified number of times?” Despite many efforts, this question remains very open after more than a century. In this talk, I will present the conjectural scaling limit …

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Permutations in Random Geometry

Random geometry and random permutations have been extremely active fields of research for several years. The former is characterized by the study of large planar maps and their continuum limits, i.e. the Brownian map, Liouville quantum gravity surfaces and Schramm–Loewner evolutions. The latter is characterized by the study of large uniform permutations and (more recently) …

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The skew Brownian permuton: a new universal limit for random constrained permutations and its connections with Liouville quantum gravity

Consider a large random permutation satisfying some constraints or biased according to some statistics. What does it look like? In this seminar we make sense of this question introducing the notion of permuton. Permuton convergence has been established for several models of random permutations in various works: we give an overview of some of these …

The skew Brownian permuton: a new universal limit for random constrained permutations and its connections with Liouville quantum gravityRead More »

The skew Brownian permuton

Consider a large random permutation satisfying some constraints or biased according to some statistics. What does it look like? In this seminar we make sense of this question introducing the notion of permutons. Permuton convergence has been established for several models of random permutations in various works: we give an overview of some of these results, …

The skew Brownian permutonRead More »

The limiting shape of random permutations: an introduction to permuton convergence.

In this series of two lectures we overview some recent progress in the study of the liming shape of large random (non uniform) permutations.  We start by properly introducing the notion of permuton convergence and by exploring its connection with the convergence of proportion of pattern densities, this being a striking feature of the permuton topology.  In …

The limiting shape of random permutations: an introduction to permuton convergence.Read More »

The feasible region for consecutive patterns of permutations is a cycle polytope

We study proportions of consecutive occurrences of permutation patterns of a given size . Specifically, the feasible limits of such proportions on large permutations form a region, called feasible region. We show that this feasible region is a polytope, more precisely the cycle polytope of a specific graph called overlap graph. The latter is a …

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