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 investigated in the last decade. In the first part of the talk, we introduce a new …
Local limits for permutations and generating treesRead More »
We look at geometric limits of large random non-uniform permutations. We mainly consider two theories for limits of permutations: permuton limits, introduced by Hoppen, Kohayakawa, Moreira, Rath, and Sampaio to define a notion of scaling limits for permutations; and Benjamini-Schramm limits, introduced by the author to define a notion of local limits for permutations. The models of …
[Ph.D] Random Permutations: A geometric point of viewRead More »
This is the webpage for the course of Stochastik. Teacher: Ashkan Nikeghbali The lectures for this course are recorded. All the material can be found here: Stochastik 2021 – Dropbox. The live sessions (Q&A sessions and exercise classes) will be on zoom (Zoom-link). The plan is as follows: -Monady 10.15-12: Q&A session with Alejandro -Wednesday 13.00-14.45: exercise …
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 by presenting the notion of permuton convergence. Then we answer the question for different choices of random permutation models. We mainly focus on two examples of permuton convergence, introducing …
Notation: Given , we set and . We also set . The model We consider a league of teams denoted by whose initial random strengths are denoted by . In the league every team plays matches, one against each of the remaining teams . Note that there are in total matches in the league. These …
We study the feasible region for consecutive patterns of pattern-avoiding permutations. More precisely, given a family of permutations avoiding a fixed set of patterns, we study the limit of proportions of consecutive patterns on large permutations of . These limits form a region, which we call the pattern-avoiding feasible region for . We show that, …
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 following images were realized during the preparation of the article “Scaling and local limits of Baxter permutations and bipolar orientations through coalescent-walk processes”. Here you can see two uniform large Baxter permutations. We showed that they converge (in the permuton sense) to the Baxter permuton, that is a random probability measure of the unit …
This is the webpage for the course of Analysis III. Teacher: Stefan Sauter Lectures: Mo 13.00 – 14.45, Room: Y03G85; Fr 13.00 – 14.45, Room: Y03G85 Exercises: Fr 10.15 – 12.00, Room: Y27H46 UZH-Webpage of the course I’m in charge of one of the exercise sessions for this course. I will publish here one exercise …
Baxter permutations, plane bipolar orientations, and a specific family of walks in the non-negative quadrant, called tandem walks, are well-known to be related to each other through several bijections. We introduce a further new family of discrete objects, called coalescent-walk processes and we relate it to the three families mentioned above. We prove joint Benjamini–Schramm …