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 …
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 …
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 …
Square permutations are tipically rectangular (with Erik Slivken)Read More »
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, …
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 …
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 …
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 …
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 …
Existence of phase transition for the level-set percolation for the discrete Gaussian free field on (DGFF) is a problem that received much attention in the past years, in particular it was studied in the 80’s by J. Bricmont, J.L. Lebowitz and C. Maes. They showed that in three dimension the DGFF has a nontrivial percolation …
Level-set percolation for the Gaussian free fieldRead More »
Nel seguente testo ci occuperemo di un argomento di estrema attualità, ancora oggi oggetto di ricerca, ovvero quello di fornire una definizione di curvatura discreta di grafi e di presentarne il calcolo in alcuni casi particolari. Il principale testo di riferimento è il recente articolo pubblicato agli inizi del 2015 in cui viene introdotta una …