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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.

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

Influence of the seed in affine preferential attachment trees
with D. Corlin Marchand,
preprint
(2018)
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,
preprint
(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,
preprint
(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,
preprint
(2018)
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,
preprint
(2017)
69 pages.

Self-avoiding walk on \(\mathbb{Z}^2\) with Yang-Baxter weights: universality of critical fugacity and 2-point function
with A. Glazman,
preprint
(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,
preprint
(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,
to appear in 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.

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).

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

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

Assistent: Corina CIOBOTARU; sous-assistent Robin Pierre KAUFMANN;

Feuilles d'exercices disponibles sur Moodle.

Mardi 15:15 - 17:00 (bâtiment de physique - auditoire 0.51) et Mercredi 15:15 - 17:00 (bâtiment de physique - auditoire 2.52)

Polycopié

Responsable: Davide Bolognini.

Rendre les copies au cours du lundi.

Semaine 3; Semaine 4; Semaine 5; Semaine 6; Semaine 7; Semaine 8; Semaine 9

Questionnaire: que pensez vous de la première partie du cours? Completer ici , a rendre le 6 novembre.

Mardi 15:15 - 17:00 et Mercredi 13:15 - 15:00 dans le bâtiment de physique - auditoires 0.51 et 2.52, respectivement

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