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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 …

[7] Scaling and local limits of Baxter permutations and bipolar orientations through coalescent-walk processes (with Mickaël Maazoun). Annals of Probability (2022), no. 50, pp. 1359–1417.Read More »

We begin with the definition of generating tree.  Since we are interested in the study of permutations, we restrict the definition to these specific objects. We need the following preliminary construction. Definition. Given a permutation and an integer we denote by the permutation obtained from by appending a new final value equal to and shifting …

[6] Asymptotic normality of consecutive patterns in permutations encoded by generating trees with one-dimensional labels. Random Structures and Algorithms (2021), no. 59, pp. 339–375.Read More »

A record in a permutation is a maximum or a minimum, from the left or from the right. The entries of a permutation can be partitioned into two types: the ones that are records are called external points, the others are called internal points. Permutations without internal points have been studied under the name of …

[5] Almost square permutations are typically square (with Enrica Duchi and Erik Slivken). Annales de l’Institut Henri Poincaré – Probab. Statist. 57 (2021), no. 4, pp. 1834-1856.Read More »

We denote the proportion of consecutive patterns in a permutation as We consider  the consecutive pattern limiting sets, called the feasible region for consecutive patterns, defined for every We are able to obtain a full description of the feasible region as the cycle polytope of a specific graph, called the overlap graph . Definition: The …

[4] The feasible region for consecutive patterns of permutations is a cycle polytope (with Raul Penaguiao). Algebraic Combinatorics 3 (2020), no. 6, pp. 1259–1281.Read More »

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 …

[3] A decorated tree approach to random permutations in substitution-closed classes (with Mathilde Bouvel, Valentin Féray, and Benedikt Stufler). Electronic Journal of Probability 25 (2020), no. 67, pp. 1–52.Read More »

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 an example and a counterexample: The left permutation …

[2] Square permutations are typically rectangular (with Erik Slivken). Annals of Applied Probability 30 (2020), no. 5, pp. 2196–2233.Read More »

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 …

[1] Local convergence for permutations and local limits for uniform ρ-avoiding permutations with |ρ|=3. Probability Theory and Related Fields 176 (2020), no. 1-2, pp. 449–531.Read More »