Jacopo

Square permutations are typically rectangular

We describe the limit (for two topologies) of large uniform random square permutations, i.e., permutations where every point is a record.  First we describe the global behavior by showing these permutations have a permuton limit which can be described by a random rectangle.  We also explore fluctuations about this random rectangle, which we can describe through coupled Brownian motions.  Second, we …

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A decorated tree approach to random permutations in substitution-closed classes (with Mathilde Bouvel, Valentin Féray, and Benedikt Stufler)

We establish a novel bijective encoding that represents permutations as forests of decorated (or enriched) trees. This allows us to prove local convergence of uniform random permutations from substitution-closed classes satisfying a criticality constraint. It also enables us to reprove and strengthen permuton limits for these classes in a new way, that uses a semi-local …

A decorated tree approach to random permutations in substitution-closed classes (with Mathilde Bouvel, Valentin Féray, and Benedikt Stufler)Read More »

Square permutations are tipically rectangular (with Erik Slivken)

ArXiv A three-pages abstract for Permutation Patterns 2019 We can think of the records of a permutation (i.e., left-to-right or right-to-left maxima or minima) as the external points of a permutation. The points of a permutation that do not correspond to records are called internal points.  Square permutations are permutations with no internal points. Here …

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What is… a permuton?

How does a large random permutation behave? We will try to answer this question for different classical models of random permutations, such as uniform permutations, pattern-avoiding permutations, Mallows permutations and many others. An appropriate framework to describe the asymptotic behaviour of these pemutations is to use a quite recent notion of scaling limits for permutations, …

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Random Combinatorial Structures 2019

This is the webpage for the course of Random Combinatorial Structures. Teacher: Valentin Féray Lectures: Fr 10.15 – 12.00, Room: Y27H25 Exercises: Mo 8.25 – 10.00, Room: Y27H25 UZH-Webpage of the course Some notes on the various notions of convergence for real-valued random variables: CONVERGENCE FOR RANDOM VARIABLES I’m in charge of the exercise sessions …

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Enumerative Combinatorics 2018

This is the webpage for the course of Enumerative Combinatorics. Teacher: Mathilde Bouvel Lectures: Th 15.00 – 17.00, Room: Y27H28 Exercises: Th 8.00 – 9.45, Room: Y27H46 UZH-Webpage of the course I’m in charge of the exercise sessions for this course. I will publish here one exercise sheet per week. Exercises 1 (Solutions 1) Exercises …

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Local convergence for permutations and local limits for uniform ρ-avoiding permutations with |ρ|=3. Probability Theory and Related Fields, to appear.

We set up a new notion of local convergence for permutations and we prove a characterization in terms of proportions of consecutive pattern occurrences. We also characterize random limiting objects for this new topology introducing a notion of “shift-invariant” property (corresponding to the notion of unimodularity for random graphs). We then study two models in …

Local convergence for permutations and local limits for uniform ρ-avoiding permutations with |ρ|=3. Probability Theory and Related Fields, to appear.Read More »

Local convergence for random permutations: the case of uniform pattern-avoiding permutations.

For large combinatorial structures, two main notions of convergence can be defined: scaling limits and local limits. In particular for graphs, both notions are well-studied and well-understood. For permutations only a notion of scaling limits, called permutons, has been recently introduced. The convergence for permutons has also been characterized by frequencies of pattern occurrences. We …

Local convergence for random permutations: the case of uniform pattern-avoiding permutations.Read More »