Office 1.102 (1st floor)
+41 26 300 9533
Departement de mathematiques
Université de Fribourg
Chemin du Musée 23
CH-1700 Fribourg, Switzerland


I am researcher in mathematics, currently working a professor the University of Fribourg. My research interests lie in probability, more precisely in problems inspired by statistical mechanics. I specifically work with percolation, the random-cluster and Potts models, and self-avoiding walk.

Previously, I was a student of the ENS Paris and have obtained my PhD from the University of Cambridge under the supervision of Geoffrey Grimmett in 2012. From 2012 to 2015 I was a postdoc at the University of Geneva, in the group of Stanislav Smirnov and Hugo Duminil-Copin. A more detailed CV is available here.


Available Ph.D. position in statistical mechanics from the 1st of September 2018. Click here for details.

Past News:

The 2017 Plancherel Lecture will be held by Hugo Duminil-Copin on October 9th. Click here for details.

Publications & other material


You may also look at my arXiv or google scholar pages.

Universality for the random-cluster model on isoradial graphs with H. Duminil-Copin and J.H Li preprint (2017) 69 pages.
Abstract. We show that the canonical random-cluster measure associated to isoradial graphs is critical for all \(q \geq 1\). Additionally, we prove that the phase transition of the model is of the same type on all isoradial graphs: continuous for \(1 \leq q \leq 4\) and discontinuous for \(q > 4\). For \(1 \leq q \leq 4\), the arm exponents (assuming their existence) are shown to be the same for all isoradial graphs. In particular, these properties also hold on the triangular and hexagonal lattices. Our results also include the limiting case of quantum random-cluster models in \(1+1\) dimensions.
Abstract. We consider a self-avoiding walk model (SAW) on the faces of the square lattice \(\mathbb{Z}^2\). This walk can traverse the same face twice, but crosses any edge at most once. The weight of a walk is a product of local weights: each square visited by the walk yields a weight that depends on the way the walk passes through it. The local weights are parametrised by angles \(\theta\in[\frac{\pi}{3},\frac{2\pi}{3}]\) and satisfy the Yang--Baxter equation. The self-avoiding walk is embedded in the plane by replacing the square faces of the grid with rhombi with corresponding angles. By means of the Yang-Baxter transformation, we show that the 2-point function of the walk in the half-plane does not depend on the rhombic tiling (i.e. on the angles chosen). In particular, this statistic coincides with that of the self-avoiding walk on the hexagonal lattice. Indeed, the latter can be obtained by choosing all angles \(\theta\) equal to \(\frac{\pi}{3}\). For the hexagonal lattice, the critical fugacity of SAW was recently proved to be equal to \(1+\sqrt{2}\). We show that the same is true for any choice of angles. In doing so, we also give a new short proof to the fact that the partition function of self-avoiding bridges in a strip of the hexagonal lattice tends to 0 as the width of the strip tends to infinity. This proof also yields a quantitative bound on the convergence.
Discontinuity of the phase transition for the planar random-cluster and Potts models with \(q>4\) with H. Duminil-Copin, M. Gagnebin, M. Harel, V. Tassion, preprint (2016) 43 pages.
Abstract. We prove that the \(q\)-state Potts model and the random-cluster model with cluster weight \(q>4\) undergo a discontinuous phase transition on the square lattice. More precisely, we show
- Existence of multiple infinite-volume measures for the critical Potts and random-cluster models,
- Ordering for the measures with monochromatic (resp. wired) boundary conditions for the critical Potts model (resp. random-cluster model), and
- Exponential decay of correlations for the measure with free boundary conditions for both the critical Potts and random-cluster models.
The proof is based on a rigorous computation of the Perron-Frobenius eigenvalues of the diagonal blocks of the transfer matrix of the six-vertex model, whose ratios are then related to the correlation length of the random-cluster model.
As a byproduct, we rigorously compute the correlation lengths of the critical random-cluster and Potts models, and show that they behave as \(\exp(\pi^2/\sqrt{q-4})\) as \(q\) tends to 4.
The Bethe ansatz for the six-vertex and XXZ models: an exposition with H. Duminil-Copin, M. Gagnebin, M. Harel, V. Tassion, preprint (2016) 22 pages.
Abstract. In this paper, we review a few known facts on the coordinate Bethe ansatz. We present a detailed construction of the Bethe ansatz vector \(\psi\) and energy \(\Lambda\), which satisfy \(V \psi = \Lambda \psi\), where \(V\) is the the transfer matrix of the six-vertex model on a finite square lattice with periodic boundary conditions for weights \(a= b=1\) and \(c > 0\). We also show that the same vector \(\psi\) satisfies \(H \psi = E \psi\), where \(H\) is the Hamiltonian of the XXZ model (which is the model for which the Bethe ansatz was first developed), with a value \(E\) computed explicitly.
Variants of this approach have become central techniques for the study of exactly solvable statistical mechanics models in both the physics and mathematics communities. Our aim in this paper is to provide a pedagogically-minded exposition of this construction, aimed at a mathematical audience. It also provides the opportunity to introduce the notation and framework which will be used in a subsequent paper by the authors that amounts to proving that the random cluster model on \(\mathbb{Z}^2\) with cluster weight \(q >4\) exhibits a first-order phase transition.
Expected depth of random walks on groups with K. Bou-Rabee and A. Myropolska, preprint (2016) 14 pages.
Abstract. For \(G\) a finitely generated group and \(g \in G\), we say \(g\) is detected by a normal subgroup \(N \lhd G\) if \(g \notin N\). The depth \(D_G(g)\) of \(g\) is the lowest index of a normal, finite index subgroup \(N\) that detects \(g\). In this paper we study the expected depth, \(\mathbb{E}[D_G(X_n)]\), where \(X_n\) is a random walk on \(G\). We give several criteria that imply that \[ \mathbb{E}[D_G(X_n)] \xrightarrow[n\to \infty]{} 2 + \sum_{k \geq 2}\frac{1}{[G:\Lambda_k]}\,,\] where \(\Lambda_k\) is the intersection of all normal subgroups of index at most \(k\). We explain how the right-hand side above appears as a natural limit and also give an example where the convergence does not hold.
Abstract. We prove sharpness of the phase transition for the random-cluster model with \(q \geq 1\) on graphs of the form \( S := G \times S \), where \(G\) is a planar lattice with mild symmetry assumptions, and \(S\) a finite graph. That is, for any such graph and any \(q \geq 1\), there exists some parameter \(p_c = p_c(S,q)\), below which the model exhibits exponential decay and above which there exists a.s. an infinite cluster.
The proof adapts the methods of [6] to non-planar graphs using techniques developed in [7]. The result is also valid for the random-cluster model on planar graphs with long range, compactly supported interaction. It extends to the Potts model via the Edwards- Sokal coupling.
The phase transitions of the planar random-cluster and Potts models with \(q \geq 1\) are sharp with H. Duminil-Copin, Probability Theory and Related Fields (2016) 23 pages, 4 figures.
Abstract. We prove that random-cluster models with \( q \geq 1 \) on a variety of planar lattices have a sharp phase transition, that is that there exists some parameter pc below which the model exhibits exponential decay and above which there exists a.s. an infinite cluster. The result may be extended to the Potts model via the Edwards-Sokal coupling.
Our method is based on sharp threshold techniques and certain symmetries of the lattice; in particular it makes no use of self-duality. Part of the argument is not restricted to planar models and may be of some interest for the understanding of random-cluster and Potts models in higher dimensions.
Due to its nature, this strategy could be useful in studying other planar models satisfying the FKG lattice condition and some additional differential inequalities.
Scaling limits and influence of the seed graph in preferential attachment trees with N. Curien, T. Duquesne and I. Kortchemski, Journal de l'Ecole Polytechnique (2015) 33 pages, 11 figures.
Abstract. We are interested in the asymptotics of random trees built by linear preferential attachment, also known in the literature as Barabasi-Albert trees or plane-oriented recursive trees. We first prove a conjecture of Bubeck, Mossel & Racz concerning the influence of the seed graph on the asymptotic behavior of such trees. Separately we study the geometric structure of nodes of large degrees in a plane version of Barabasi-Albert trees via their associated looptrees. As the number of nodes grows, we show that these looptrees, appropriately rescaled, converge in the Gromov-Hausdorff sense towards a random compact metric space which we call the Brownian looptree. The latter is constructed as a quotient space of Aldous' Brownian Continuum Random Tree and is shown to have almost sure Hausdorff dimension 2.
Planar lattices do not recover from forest fires with D. Kiss and V. Sidoravicius, Annals of Probability (2015) 24 pages, 6 figures.
Abstract. Self-destructive percolation with parameters \(p,\delta\) is obtained by taking a site percolation configuration with parameter \(p\), closing all sites belonging to infinite clusters, then opening every closed site with probability \(\delta\), independently of the rest. Call \(\theta(p,\delta)\) the probability that the origin is in an infinite cluster in the configuration thus obtained.
For two dimensional lattices, we show the existence of \(\delta > 0\)such that, for any \(p > p_c\), \(\theta(p,\delta) = 0\). This proves the conjecture of van den Berg and Brouwer, who introduced the model. Our results combined with those of van den Berg and Brouwer imply the non-existence of the infinite parameter forest-fire model. The methods herein apply to site and bond percolation on any two dimensional planar lattice with sufficient symmetry.
On the probability that self-avoiding walk ends at a given point with H. Duminil-Copin, A. Glazman and A. Hammond, Annals of Probability (2016) 30 pages, 7 figures.
Abstract. We prove two results on the delocalization of the endpoint of a uniform self-avoiding walk on \(\mathbb{Z}^d\) for \(d \geq 2\). We show that the probability that a walk of length \(n\) ends at a point \(x\) tends to \(0\) as \(n\) tends to infinity, uniformly in \(x\). Also, when \(x\) is fixed, with \(\vert\vert x \vert\vert = 1\), this probability decreases faster than \(n^{-1/4 + \epsilon}\) for any positive \(\epsilon\).This provides a bound on the probability that a self-avoiding walk is a polygon.
Bond percolation on isoradial graphs: criticality and universality with G. Grimmett, Probability Theory and Related Fields (2013) 63 pages, 24 figures.
Abstract. In an investigation of percolation on isoradial graphs, we prove the criticality of canonical bond percolation on isoradial embeddings of planar graphs, thus extending celebrated earlier results for homogeneous and inhomogeneous square, triangular, and other lattices. This is achieved via the star-triangle trasformation, by transporting the box-crossing property across the family of isoradial graphs. As a consequence, we obtain the universality of these models at the critical point, in the sense that the one-arm and 2j-alternating-arm critical exponents (and therefore also the connectivity and volume exponents) are constant across the family of such percolation processes. The isoradial graphs in question are those that satisfy certain weak conditions on their embedding and on their track system. This class of graphs includes, for example, isoradial embeddings of periodic graphs, and graphs derived from rhombic Penrose tilings.
Universality for bond percolation in two dimensions with G. Grimmett, Annals of Probability (2013) 25 pages, 9 figures.
Abstract. All (in)homogeneous bond percolation models on the square, triangular, and hexagonal lattices belong to the same universality class, in the sense that they have identical critical exponents at the critical point (assuming the exponents exist). This is proved using the star-triangle transformation and the box-crossing property. The exponents in question are the one-arm exponent \(\rho\), the \(2j\)-alternating-arms exponents \(\rho_{2j}\) for \(j \ge 1\), the volume exponent \(\deltalta\), and the connectivity exponent \(\eta\). By earlier results of Kesten, this implies universality also for the near-critical exponents \(\beta\), \(\gamma\), \(\nu\), \(\deltalta\) (assuming these exist) for any of these models that satisfy a certain additional hypothesis, such as the homogeneous bond percolation models on these three lattices.
Inhomogeneous bond percolation on square, triangular and hexagonal lattices with G. Grimmett, Annals of Probability (2013) 36 pages, 15 figures.
Abstract. The star-triangle transformation is used to obtain an equivalence extending over the set of all (in)homogeneous bond percolation models on the square, triangular, and hexagonal lattices. Amongst the consequences are box-crossing (RSW) inequalities for such models with parameter-values at which the transformation is valid. This is a step towards proving the universality and conformality of these processes. It implies criticality of such values, thereby providing a new proof of the critical point of inhomogeneous systems. The proofs extend to certain isoradial models to which previous methods do not apply.


Universality for planar percolation under the supervision of G. Grimmett (2012).


Enseignement 2017/2018

Algèbre linéaire propédeutique (semestre d'automne)

Cours (MA 0201):
Lundi 13:15 - 15:00 dans le bâtiment de physiologie - grand auditoire 1.100
Polycopié - version abrégée - version complète
Exercices (MA 0261):
Jeudi 8:15 - 9:00 dans le bâtiment de chimie - grand auditoire 0.013
Responsable: Linard HOESSLY
Feuilles d'exercices disponibles sur Moodle avec la clé d'incription "algebra".
Tablaux: semaine 1; semaine 2; semaine 3; semaine 4; semaine 5; semaine 6; semaine 7; semaine 8; semaine 9; semaine 10; semaine 11; semaine 12; semaine 13; semaine 14; sujet examen sept 2017

Probabilités et statistique (semestre d'automne)

Cours (MA.2431):
Jeudi 13:15-15:00 dans le bâtiment de physique - auditoire 2.52
Polycopié, version préliminaire
Vendredi 10:15 - 12:00 dans le bâtiment de physique - auditoire 2.52
Responsable: Xavier RICHARD
Feuilles d'exercices disponibles sur Moodle.
Une video sur l'ordre de grandeur de 52! sur YouTube.

Mesure et intégration (semestre d'automne)

Cours (MA.3401/3402 BSc; MA.4401/4402 MSc):
Lundi 15:15 - 17:00 et Mardi 15:15 - 17:00 dans le bâtiment de physique - auditoire 0.51
Polycopié, version préliminaire
Responsable: Basil Reinhard.
Rendre dans la boitte aux lettres au bâtiment de physique marquée "Mesure et Intégration" le vendredi avant 13h.
Heures de consultations (office hours), le jeudi de 9h à 10h, Lonza bureau 0.105 ou sur rendez-vous par mail à
Serie 1, Serie 2, Serie 3, Serie 4, Serie 5, Serie 6, Serie 7, Serie 8, Serie 10, Serie 12

Probabilités (semestre de printemps)

Cours (MA.3412; MA.4412):
Mardi 13:15 - 15:00 et Mercredi 15:15 - 17:00 dans le bâtiment de physique - auditoire 2.52
Le cours suit le livre Probability: Theory and Examples par Rick Durrett et les seconde et troisieme parties des notes de cours de J.F Le Gall

Le cours Mesure et intégration est fortement conseillé comme prérequis
Office 1.102 (1st floor)
+41 26 300 9533
Departement de mathematiques
Université de Fribourg
Chemin du Musée 23
CH-1700 Fribourg, Switzerland