# Vincent Beffara

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 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. I am spending the academic year 2013-2014 at the Max-Planck Institut in Bonn, Germany.

### Brownian motion

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 processes

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).

Hausdorff dimensions for \(\mathrm{SLE}_6\).Ann. Probab. 32 (2004), no. 3B, 2606-2629. PDF.

The dimension of the SLE curves.Ann. Probab. 36 (2008), no. 4, 1421-1452. PDF.

### 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.With A. Bálint and V. Tassion.ALEA 10 (2013), pp. 653-666.

Confidence intervals for the critical value in the divide and color model.With A. Bálint and V. Tassion.ALEA 10 (2013), pp. 667-679.

### 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.Probability Theory and Related Fields 147 (2010), pp. 565-581. PDF.

### Self-interacting random walks

Scaling limit of the prudent walk.With S. Friedli and Y. Velenik.Electron. Commun. Probab. 15 (2010), pp. 44-58.

### 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).

Mouvement brownien plan, SLE, invariance conforme et dimensions fractales.PhD thesis, Université Paris-Sud (Orsay). English, introduction in French. Full text, Introduction.

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.Gazette des Mathématiciens 128 (2011), pp. 5-14. PDF.

## Other stuff

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.

### Critical values

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 ...

### Elsewhere

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 !