Office 1.102 (1st floor)

+41 26 300 9533

Université de Fribourg

Chemin du Musée 23

CH-1700 Fribourg, Switzerland

I am researcher in mathematics, currently working as a professor at 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.

**Le programme Leonardo commence en septembre 2021!**

Il permet aux collégiennes et collégiens de suivre des cours de mathématiques à l'Université de Fribourg.

The **meeting of the of the Swiss Mathematical Society** titled
**Recent advances in loop models and height functions** was held at the University of Fribourg from the 2nd to the 4th of Sept. 2019.

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

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

We answer the following question: if the occupied (or vacant) set of a planar Poisson Boolean percolation model does contain a crossing of an n x n square, how wide is this crossing? The answer depends on the whether we consider the critical, sub- or supercritical regime, and is different for the occupied and vacant sets.

Near critical scaling relations for planar Bernoulli percolation without differential inequalities
with H. Duminil-Copin and V. Tassion
Preprint
(2021)
19 pages.

We provide a new proof of the near-critical scaling relation \(\beta=\xi_1\nu\) for Bernoulli percolation on the square lattice already proved by Kesten in 1987. We rely on a novel approach that does not invoke Russo's formula, but rather relates differences in crossing probabilities at different scales. The argument is shorter and more robust than previous ones and is more likely to be adapted to other models. The same approach may be used to prove the other scaling relations appearing in Kesten's work.

Structure of Gibbs measures for planar FK-percolation and Potts models
with A. Glazman
to appear in Probability and Mathematical Physics
(2021)
46 pages.

Most of the work is devoted to the FK-percolation model on \(Z^2\) with \(q \geq 1\), where we prove that every Gibbs measure is a linear combination of the free and wired ones. The arguments are non-quantitative and follow the spirit of the seminal works of Aizenman and Higuchi, which established the Gibbs structure for the two-dimensional Ising model. Infinite-range dependencies in FK-percolation (i.e., a weaker spatial Markov property) pose serious additional difficulties compared to the case of the Ising model. For example, it is not automatic, albeit true, that thermodynamic limits are Gibbs. The result for the Potts model is then derived using the Edwardsâ€“Sokal coupling and auto-duality. The latter ingredient is necessary since applying the Edwardsâ€“Sokal procedure to a Gibbs measure for the Potts model does not automatically produce a Gibbs measure for FK-percolation.

Finally, the proof is generic enough to adapt to the FK-percolation and Potts models on the triangular and hexagonal lattices and to the loop O(n) model in the range of parameters for which its spin representation is positively associated.

Delocalization of the height function of the six-vertex model
with H. Duminil-Copin, A. Karrila, M. Oulamara
Preprint
(2020)
54 pages.

Rotational invariance in critical planar lattice models
with H. Duminil-Copin, K.K. Kozlowski, D. Krachun, M. Oulamara
Preprint
(2020)
91 pages.

On the six-vertex model's free energy
with H. Duminil-Copin, K.K. Kozlowski, D. Krachun, T. Tikhonovskaia
Communications in Mathematical Physics
(2022)
52 pages.

Planar random-cluster model: scaling relations
with H. Duminil-Copin,
To appear in Forum of Mathematics, Pi
(2020)
85 pages.

Planar random-cluster model: fractal properties of the critical phase
with H. Duminil-Copin and V. Tassion,
Probability Theory and Related Fields
(2021)
40 pages.

We also obtain a number of properties of so-called arm-events: three universal critical exponents (two arms in the half-plane, three arms in the half-plane and five arms in the bulk), quasi-multiplicativity and well-separation properties (even when arms are not alternating between primal and dual), and the fact that the four-arm exponent is strictly smaller than 2. These results were previously known only for Bernoulli percolation (\(q = 1\)) and the FK-Ising model (\(q = 2\)).

Finally, we prove new bounds on the one, two and four arms exponents for \(q\in[1,2]\). These improve the previously known bounds, even for Bernoulli percolation.

Influence of the seed in affine preferential attachment trees
with D. Corlin Marchand,
Bernoulli Journal
(2020)
31 pages.

We study the problem of the asymptotic influence of the seed \(S\) on the law of \(T_n^S\). We show that, for any two distinct seeds \(S\) and \(S'\), the laws of \(T_n^S\) and \(T_n^{S'}\) remain at uniformly positive total-variation distance as \(n\) increases.

This is a continuation of Curien et al. (2015), which in turn was inspired by a conjecture of Bubeck et al. (2015). The technique developed here is more robust than previous ones and is likely to help in the study of more general attachment mechanisms.

Exponential decay in the loop \(O(n)\) model: \(n> 1\), \(x<\tfrac{1}{\sqrt{3}}+\varepsilon(n)\)
with A. Glazman,
Special volume in honour of Vladas Sidoravicius -- Progress in Probability
(2018)
12 pages.

It is expected that, for \(n \leq 2\), the model exhibits a phase transition in terms of \(x\), that separates regimes of polynomial and exponential decay of loop sizes. In this paradigm, our result implies that the phase transition for \(n \in (1,2]\) occurs at some critical parameter \(x_c(n)\) strictly greater than that \(x_c(1) = 1/\sqrt3\). The value of the latter is known since the loop \(O(1)\) model on the hexagonal lattice represents the contours of spin-clusters of the Ising model on the triangular lattice.

The proof is based on developing \(n\) as \(1+(n-1)\) and exploiting the fact that, when \(x<\tfrac{1}{\sqrt{3}}\), the Ising model exhibits exponential decay on any (possibly non simply-connected) domain. The latter follows from the positive association of the FK-Ising representation.

Uniform Lipschitz functions on the triangular lattice have logarithmic variations
with A. Glazman,
Communications in Mathematical Physics
(2018)
64 pages.

The level lines of such functions form a loop \(O(2)\) model on the edges of the hexagonal lattice with edge-weight one. An infinite-volume Gibbs measure for the loop \(O(2)\) model is constructed as a thermodynamic limit and is shown to be unique. It contains only finite loops and has properties indicative of scale-invariance: macroscopic loops appearing at every scale. The existence of the infinite-volume measure carries over to height functions pinned at 0; the uniqueness of the Gibbs measure does not.

The proof is based on a representation of the loop \(O(2)\) model via a pair of spin configurations that are shown to satisfy the FKG inequality. We prove RSW-type estimates for a certain connectivity notion in the aforementioned spin model.

Bounding the number of self-avoiding walks: Hammersley-Welsh with polygon insertion
with H. Duminil-Copin, S. Gangulyand and A. Hammond,
Annals of Probability
(2020)
54 pages.

We consider two planar lattices and prove that \(c_n \leq \exp(C n^{1/2 -\epsilon}) \mu^n\) for an explicit constant \(\epsilon> 0\) (where here \(\mu\) denotes the connective constant for the lattice in question). The result is conditional on a lower bound on the number of self-avoiding polygons of length \(n\), which is proved to hold on the hexagonal lattice \(\mathbb H\) for all \(n\), and subsequentially in \(n\) for \(\mathbb Z^2\). A power-law upper bound on \(c_n \mu^{-n}\) for \(\mathbb H\) is also proved, contingent on a non-quantitative assertion concerning this lattice's connective constant.

Universality for the random-cluster model on isoradial graphs
with H. Duminil-Copin and J.H Li,
Electronic Journal of Probability
(2018)
69 pages.

Self-avoiding walk on \(\mathbb{Z}^2\) with Yang-Baxter weights: universality of critical fugacity and 2-point function
with A. Glazman,
Annales de l'Institut Henri Poincaré
(2017)
24 pages.

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,
to appear in Annales Scientifiques de l'École Normale Supérieure
(2016)
43 pages.

- 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,
Probability Surveys
(2018)
22 pages.

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,
Pacific Journal of Mathematics
(2016)
14 pages.

The phase transitions of the random-cluster and Potts models on slabs with \(q \geq 1\) are sharp
with A. Raoufi,
Electronic Journal of Probability
(2018)
24 pages, 6 figures.

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.

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.

Planar lattices do not recover from forest fires
with D. Kiss and V. Sidoravicius,
Annals of Probability
(2015)
24 pages, 6 figures.

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.

Bond percolation on isoradial graphs: criticality and universality
with G. Grimmett,
Probability Theory and Related Fields
(2013)
63 pages, 24 figures.

Universality for bond percolation in two dimensions
with G. Grimmett,
Annals of Probability
(2013)
25 pages, 9 figures.

Inhomogeneous bond percolation on square, triangular and hexagonal lattices
with G. Grimmett,
Annals of Probability
(2013)
36 pages, 15 figures.

Uniform Lipschitz functions on hexagonal lattice
talk given at "Probability and quantum field theory: discrete models, CFT, SLE and constructive aspects" (Porquerolles)
(2019).

First order phase transition for the Random Cluster model with q > 4 talk given at Diablerets conference
(2017).

Scaling limits and influence of the seed graph in preferential attachment trees talk given at IHES
(2014).

Planar lattices do not recover from forest fires talk given at Tel Aviv university
(2014).

Delocalization of the endpoint of self-avoiding walk talk given at the ENS Lyon
(2013).

Percolation on isoradial graphs talk given at IMPA, Brazil
(2013).

First order phase transition for the Random Cluster model with q > 4

Scaling limits and influence of the seed graph in preferential attachment trees

Planar lattices do not recover from forest fires

Delocalization of the endpoint of self-avoiding walk

Percolation on isoradial graphs

Lundi 13:15 - 15:00 dans le bâtiment de chimie (PER 10), grand auditoire 0.013

Polycopié - version abrégée - version complète

Jeudi 8:15 - 9:00 dans le bâtiment de chimie (PER 10), grand auditoire 0.013

Responsable: Xavier Richard

Feuilles d'exercices disponibles sur Moodle.

La page pour le cours est accessible ici sans mot de passe.

Vendredi 10:15 - 12:00 dans le bâtiment de physique (Per 08) - auditoire 2.52

Polycopié, version préliminaire

Jeudi 13:15-15:00 dans le bâtiment de physique (Per 08) - auditoire 2.52

Feuilles d'exercices disponibles sur Moodle.

Une video sur l'ordre de grandeur de 52! sur YouTube.

Mardi 15:15 - 17:00 et Mercredi 15:15 - 17:00 dans le bâtiment de physique - auditoires 0.51 et 2.52, respectivement