Webpage of Armand Riera

ARMAND RIERA

Associate Professor (MCF) at Sorbonne université | Research Team : Stochastic analysi and random structures.

My research interests lie primarily on the field of 2-dimensional random geometry, specifically in the study of scaling limits of models of random planar maps. These limit objects are used to construct natural measures on certain classes of metric spaces and allow to define random surfaces. This theory, commonly referred to as Brownian Geometry, is an interdisciplinary field that combines elements of combinatorics, physics, and probability theory.

To investigate these structures, I use a range of stochastic processes including growth-fragmentation processes, branching processes, and superprocesses.

Research

The scaling limit of planar maps with large faces
| with Nicolas Curien and Grégory Miermont (To appear, 186 pages)
- We prove that large Boltzmann stable planar maps of index $\alpha \in (1;2)$ converge in the scaling limit towards a random compact metric space $ \mathcal{S}_{\alpha}$ that we construct explicitly. They form a one-parameter family of random continuous spaces ``with holes'' or ``faces'' different from the Brownian sphere. In the so-called dilute phase $ \alpha \in [3/2;2)$, the topology of $ \mathcal{S}_{\alpha}$ is that of the Sierpinski carpet, while in the dense phase $ \alpha \in (1;3/2)$ the ``faces'' of $ \mathcal{S}_{\alpha}$ may touch each-others. En route, we prove various geometric properties of these objects concerning their faces or the behavior of geodesics.
  
  ArXiv version. Video.
Excursion theory for Markov processes indexed by Lévy trees
| with Alejandro Rosales-Ortiz (To appear, 64 pages)
Self-similar Markov trees and scaling limits
| with Jean Bertoin and Nicolas Curien (To appear in IMS Monographs, 264 pages)
Peeling the Brownian half-plane
| with Jean-François Le Gall (To appear in Probab. Th. Rel. Fields, 28 pages)
Spatial Markov property in Brownian disks
| with Jean-François Le Gall (To appear in Ann. Inst. Henri Poincaré Probab. Stat. 43 pages)
The structure of the local time of Markov processes indexed by Levy trees
| with Alejandro Rosales-Ortiz (Probab. Th. Rel. Fields 189 (2024), 99 pages)
Isoperimetric inequalities in the Brownian plane
| (Ann. Probab. 50 (2022), 42 pages)
Spine representations for non-compact models of random geometry
| with Jean-François Le Gall (Probab. Th. Rel. Fields 181 (2021), 74 pages)
Some explicit distributions for Brownian motion indexed by the Brownian tree
| with Jean-François Le Gall (Markov Processes Relat. Fields 26 (2020), 27 pages, Special issue)
Growth-fragmentation processes in Brownian motion indexed by the Brownian tree
| with Jean-François Le Gall (Ann. Probab. 48 (2020), 42 pages)
PhD Thesis: Brownian Geometry.
| Under the supervision of Jean-François Le Gall. This thesis was awarded the Prix de thèse Jacques Neveu 2022 de la SMAI.

Surveys and proceedings

Surfaces Browniennes. Disclosure article in French for the Neveu prize (Matapli Journal. 130 (2023), 18 pages).

Report of the paper The scaling limit of planar maps with large faces. (Oberwolfach Reports. 130 (2025), 5 pages).

Student supervision

PhD:
Pablo Zúñiga: PhD (co-supervision with Nicolas Broutin) starting in 2025.

Fael Rebei: Pre-PhD 2024-2025 and PhD starting in 2025 (co-supervision with Thomas Duquesne).

Alejandro Rosales-Ortiz : 2021-2023 (co-supervision with Jean Bertoin).
From Step Reinforced Random Walks to Noise Reinforced Lévy Processes and Excursion Theory for Markov Processes Indexed by Lévy Trees.
This thesis was awarded with a PhD Distinction from the University of Zurich.

Master:
Sammy Chabni : 2025 ; Pablo Zúñiga Rodríguez-Peña : 2025 ; Corentin Gratin : 2025 ; Fael Rebei : 2024 (co-supervision with Thomas Duquesne) ; Zheng Fang : 2022 (co-supervision with Jean Bertoin) ; Rebekka Rakutt : 2022 (co-supervision with Jean Bertoin).

Miscellaneous

Course : Statistiques des processus M1 ISUP.

Exercises sessions for Nicolas Curien's Courses: Random surfaces at Saint-Flour 2019 and Self-Similar Markov Branching Trees at Mexico 2022.

Master thesis M2: Étude des limites de permutations aléatoires et application aux permutations séparables.

Surveys and proceedings

Student supervision

Miscellaneous

Curriculum Vitae