Welcome to my web page. Here, you will find a description of my past and current research, with links to the corresponding publications. Things like images and such are on separate pages, see below.
Research and publications
I am currently a chargé de recherche at the math department of ENS Lyon, and a member of the probability team. My main research interest is in statistical physics, especially the study of critical phenomena in two dimensions.
My first paper was about the existence of some exceptional points on the typical two-dimensional Brownian motion path, called pivoting point: That is, cut-points around which one half of the path can rotate of a positive angle without intersecting the other half. Such points always exist for sufficiently small angles - even though they are all winding points …
On conformally invariant subsets of the planar Brownian curve. Ann. Inst. H. Poincaré Probab. Statist. 39 (2003), no. 5, 793-821. PDF.
SLE is a stochastic process in two dimensions introduced by Oded Schramm as a candidate to be the scaling limit of various critical models (such as loop-erased random walks, self-avoiding walks, percolation cluster boundaries etc.). My main result on the subject is the derivation of the Hausdorff dimension of the trace of the process: First for the parameter \(\kappa=6\) (first paper), then for all \(\kappa\neq4\) (version in my thesis), then for all \(\kappa\ge0\) (second paper).
Percolation and random-cluster models
I am working on various percolation-related models in two dimensions, at criticality. The main question I am inerested in is that of conformal invariance, but so far the proofs are restricted to very specific models (work of Smirnov). The main paper in this list is most likely the 4th one.
Cardy's formula on the triangular lattice, the easy way. Universality and Renormalization, vol. 50 of the Fields Institute Communications (2007), pp. 39-45. PDF.
Is critical 2D percolation universal? In and Out of Equilibrium 2 (2008), vol. 60 of Progress in Probability, Birkhäuser, pp. 31-58. PDF.
On monochromatic arm exponents for critical 2D percolation. With P. Nolin. Annals of Probability 40 (2012), pp. 1286-1304. Preprint arXiv:0906.3570. PDF. This one has a numerical companion paper with estimates for the monochromatic exponents. Please have a look at it!
The self-dual point of the two-dimensional random-cluster model is critical for \(q\geq 1\). With H. Duminil-Copin. Probability Theory and Related Fields 153 (2012), pp. 511-542. Preprint arXiv:1006.5073. PDF, Online.
Smirnov's fermionic observable away from criticality. With H. Duminil-Copin. Annals of Probability 40 (2012), pp. 2667-2689.
The self-dual point of the two-dimensional random-cluster model is critical above \(4\). With H. Duminil-Copin and S. Smirnov. While less general than the previous one, this paper uses the parafermionic observable introduced by Smirnov to get a more "modern" proof. PDF.
On the critical value function of the DaC model, I: qualitative features. With A. Bálint and V. Tassion. To appear in ALEA (2013).
On the critical value function of the DaC model, II: numerical estimates. With A. Bálint and V. Tassion. To appear in ALEA (2013).
Last-passage percolation and interacting particle systems
This is joint work with Vladas Sidoravicius (IMPA, Rio de Janeiro), Herbert Spohn (TU-München) and Eulalia Vares (CBPF, Rio de Janeiro) on the effect of a columnar defect in two-dimensional last-passage percolation in the plane, and the relation with the so-called slow-bond problem for the one-dimensional totally asymmetric exclusion process. A second paper is currently in preparation.
Polymer pinning in a random media as influence percolation. With V. Sidoravicius, H. Spohn and E. Vares. Dynamics and Stochastics, vol. 48 of IMS Lecture Notes - Monograph series, (2006), pp. 1-15. PDF.
On a randomized PNG model with a columnar defect. With V. Sidoravicius and E. Vares. To appear in Probability Theory and Related Fields (2009). PDF.
Related things, surveys, and presentation material
Links to my PhD thesis, and to a survey article on Percolation to be part of the Encyclopedia of Mathematical Physics (Elsevier, 2006).
Mécanique Statistique et Criticalité en dimension deux. Habilitation à diriger des recherches, École Normale Supérieure de Lyon, 2011. PDF.
Percolation theory. Encyclopedia of Mathematical Physics, Elsevier, 2006. PDF.
Grands graphes planaires aléatoires et Carte brownienne. Séminaire Bourbaki 992 (2008). PDF.
SLE and other conformally invariant objects. Lecture notes for the 2010 Clay summer school on Buzios (Brazil). PDF.
La percolation, et un résultat de S. Smirnov. To appear in the Gazette des Mathématiciens (2011). PDF.
These are links to separate pages - well the titles will tell you what you will find there. The links are the same as those in the navigation block.
A few images
During my PhD and since then, I produced quite a few pictures of various two-dimensional objects, especially of SLE processes. They are located in the following page, along with a few comments and the programs used to generate them. Help yourself, have fun.
The beginning of the start of the first draft of an attempt to collect a few values of critical parameters for various models of statistical physics. But right now it is almost empty and of no particular interest to anyone. Hopefully it will grow in the future …
Just a shameless plug for a new mathematical website which we are creating, called Images des Mathématiques (yes, it's in French). It is the reincarnation of a paper publication by the French CNRS (with the same name); right now, the site only contains the old articles, but new contents are on their way.
Another one: my mother's website on old-times primary schools, which goes along with their Musée de l'école (called "il était une fois l'école"). I maintain that site, and kind of use it as a test-bed for Images des Maths, though don't tell her that !