### Selected publications

My research deals (among other things)
with lattice models coming from mathematical physics, such as
percolation-type models, spin models, self-avoiding-type walks,
random walks, random height functions, etc. My motivation to
study them is two-fold. First and more importantly, these models
are an extraordinary source of beautiful mathematical problems
(usually simple to state, but requiring new maths to solve).
Second, these models play an essential role in the understanding
of phase transition, so that one would like to study them
further to improve our understanding of the physics of critical
phenomena (conformal field theory, renormalization, exact
integrability, etc). In order to do so, I apply techniques
coming from discrete mathematics (probability, combinatorics,
graph theory, discrete analysis). In particular, I am developing
geometric tools to understand the critical behavior of a variety
of models below their critical dimension.

The list below is obviously quite a limited selection of papers I particularly like. The papers are usually related to a number of other results complementing and illustrating the potential applications. We mention some of them by numbers referring to the complete list of publications. Also, [LN]
are quite up-to-date lecture notes which may serve as an
introduction to a few of these results.

● Sharp phase transition for the random-cluster and Potts models via decision trees with A. Raoufi and V. Tassion Ann. Math. (2019).

This paper provides a general theory of sharpness of lattice spin models. The proof uses the theory of randomized algorithms coming from computer science. As a direct application, the paper proves exponential decay of correlations for two very important models of statistical physics: the random-cluster and the Potts model. This result is crucial for our understanding of the subcritical/disorder regime. Similar results were obtained in the eighties in the specific cases of Bernoulli percolation and the Ising model using BK-type correlation inequalities which are not available for general models. The technique using randomized algorithms introduced in this paper extends to a large variety of models. For instance, it allows to treat continuum percolation models such as Voronoi [40], Poisson-Boolean [50], Widom-Rowlinson [Dereudre-Houdebert], contact process, to cite but a few. It is also useful for many percolation processes obtained by taking the super-level lines of Gaussian fields; see for instance the case of the planar Bargmann-Fock process in [Muirhear-Vanneuville].

● Existence of phase transition for percolation using the Gaussian Free Field with S. Goswami, A. Raoufi, F. Severo and A. Yadin.

In this paper, we answer a conjecture raised by Benjamini and Schramm in their famous 1996 paper "Percolation beyond $\mathbb Z^d$: many questions and a few answers". Namely, we show that any Cayley graph with super-linear growth satisfies that $p_c<1$ for Bernoulli percolation (the existence of a non-trivial phase transition can be obtained for other models as well). The proof relies on a new connection between percolation and the Gaussian Free Field. This connection enables to relate the connectivity properties of percolation to geometric properties of the underlying non-necessarily transitive graph (such as isoperimetric dimension and Nash inequalities).

● Emergent Planarity in two-dimensional Ising Models with finite-range Interactions with M. Aizenman, V. Tassion and S. Warzel, to appear in

The known Pfaffian structure of the boundary spin correlations, and more generally order-disorder correlation functions, is given a new explanation through simple topological considerations within the model's random current representation. This perspective is then employed in the proof that the Pfaffian structure of boundary correlations emerges asymptotically at criticality in Ising models on $\mathbb Z^2$ with finite-range interactions. The analysis is enabled by new results on the stochastic geometry of the corresponding random currents. The proven statement establishes an aspect of universality, seen here in the emergence of fermionic structures in two dimensions beyond the solvable cases.

● Continuity of the phase transition for planar random-cluster and Potts models with $1\le q\le 4$ with V. Sidoravicius and V. Tassion, CMP (2017).

This paper develops a renormalization scheme for crossing probabilities in the critical random cluster model. Combined with the weak holomorphicity of parafermionic observables, one can prove that the phase transition of the random-cluster model with cluster-weight $q\in[1,4]$ is continuous. This answers half of a conjecture by Baxter about the behavior of the 2D Potts models. The property on crossing properties has many applications for the study of the critical model (polynomial decay of correlations, mixing properties, order of the phase transition, scaling relations). In particular, it enables one to prove that a number of mathematical definitions of a continuous phase transition are in fact equivalent. Since then, the renormalization techniques have been used in other models to prove a dichotomy result: either RSW-type estimates are valid, or there is exponential decay of correlations in the disordered phase (see e.g. [46]). The result was extended to isoradial graphs in [33].

● Discontinuity of the phase transition for the planar random-cluster and Potts models with $q>4$ with M. Gagnebin, M. Harel, I. Manolescu and V. Tassion.

This paper completes Baxter's conjecture on the behavior of the 2D Potts model by proving that the phase transition of the random-cluster model is discontinuous as soon as the cluster-weight $q$ is strictly larger than $4$. The techniques used in this paper are quite different from the one used in the companion paper (just above in the list). Here, we justified rigorously the Bethe Ansatz for the associated 6V model. Together with probabilistic techniques, this allowed us to compute the correlation length of the model explicitly at criticality, thus confirming a prediction of Baxter. The result was extended to isoradial graphs in [33].

● A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model with V. Tassion, CMP (2016).

This paper provides a completely new derivation, in three pages, of an important result in percolation theory dating from the 80', showing that probabilities of being connected decay exponentially fast in the subcritical regime. The proof extends to the Ising model. It covers the case of any transitive graph with arbitrary coupling constants. The strategy also implies that the susceptibility is infinite at criticality.

● Random currents and continuity of Ising model's spontaneous magnetization with M. Aizenman and V. Sidoravicius, CMP (2015).

This paper provides the first rigorous proof that the Ising model undergoes a continuous phase transition in 3D, and constitutes as such one of the only mathematical results on critical 3D systems. The result implies that the spin-spin correlations $\langle \sigma_0\sigma_x\rangle$ of the 3D Ising model at criticality decay slower than $C\|x\|^{-1}$ and faster than $c\|x\|^{-2}$. The proof relies on a geometric analysis of a graphical representation of the Ising model (such point of view can be used to prove a number of other results on the Ising model). The key of the proof is the introduction of a new percolation model, whose ergodic properties are related to the uniqueness of Gibbs measures at criticality; see the beautiful paper "Translation-Invariant Gibbs States of Ising model: General Setting" of Aran Raoufi for another application.

● The connective constant of the honeycomb lattice equals $\sqrt{2+\sqrt 2}$ with S. Smirnov, Ann. Math. (2012).

In 1980, the physicist Bernard Nienhuis predicted that the number $c_n$ of self-avoiding walks of length $n$ on the honeycomb lattice satisfies $c_n^{1/n}\rightarrow\sqrt{2+\sqrt 2}$ as $n$ tends to infinity. In this paper, we show this result by proving a weak form of discrete holomorphic for an observable of the model. The idea of harvesting a weak notion of discrete holomorphicity was then used in other papers dealing with self-avoiding walks, loop models, and (dependent) percolation models; see e.g. [3,6,17,19,46]. This paper represents one of the only available results in 2D (the value of the limit was conjectured by the physicist Bernard Nienhuis using the Coulomb Gas formalism in the early 80s). This result also constitutes a first step towards the proof of conformal invariance of the model on the hexagonal lattice; see [b,i]. Let us mention that understanding the typical geometric behavior of 2D self-avoiding walks is a beautiful mathematical challenge, see [12,26] for some (very partial) results and open questions.

● The self-dual point of the two-dimensional random-cluster model is critical for $q\ge1$ with V. Beffara, PTRF (2012).

In his celebrated paper of 1980, Kesten proved that the critical point of Bernoulli percolation on the square lattice $\mathbb Z^2$ is equal to $1/2$. The present paper provides the first rigorous computation of the critical point $p_c$ for an important dependent percolation model called the random-cluster model. This question goes back to the introduction of the model around 1969. The proof is based on a new Russo-Seymour-Welsh (RSW) result and sharp threshold theorems coming from the theory of boolean functions. The RSW theorem is one of the first RSW results for dependent percolation models. It was essential to the development of the theory at $p=p_c$, for instance in [30,41] (which present a more delicate description of the critical behavior). Related arguments (for instance proving the so-called sharpness of the phase transition) can be found in [24,34,38].

● The sharp threshold for bootstrap percolation in all dimensions with J. Balogh, B. Bollobás, and R. Morris, Trans. AMS (2012).

This paper shows the sharpness of the phase transition for a famous monotonic cellular automaton, called bootstrap percolation. The techniques (combining Probability Theory and Combinatorics) developed in the proof were later used to treat universality questions for a large class of monotonic automata, see e.g. [8,32,37,47,49].

● Connection probabilities and RSW-type bounds for the two-dimensional FK Ising model with C. Hongler, P. Nolin, CPAM (2011).

This paper contains the first example of a RSW-type theory for percolation models with dependency. The key is that the crossing probabilities are uniform in boundary conditions. The result was used in a variety of subsequent papers, for instance in the proof by Lubetzky and Sly that the Glauber dynamics of the critical Ising model mixes in polynomial time. Other applications can be found in [15,29]. In [23], the result was improved to cover all topological rectangles with non-degenerate extremal length. It was also instrumental in showing that the critical interfaces in the Ising and FK-Ising model on $\mathbb Z^2$ respectively converge to SLE(3) and SLE(16/3) (work [18] joint with Chelkak, Hongler, Kemppainen, and Smirnov). This latter result belongs to a short list of conformal invariance results for interfaces in 2D statistical physics.

The list below is obviously quite a limited selection of papers I particularly like. The papers are usually related to a number of other results complementing and illustrating the potential applications. We mention some of them by numbers referring to the complete list of publications. Also, [LN]

● Sharp phase transition for the random-cluster and Potts models via decision trees with A. Raoufi and V. Tassion Ann. Math. (2019).

This paper provides a general theory of sharpness of lattice spin models. The proof uses the theory of randomized algorithms coming from computer science. As a direct application, the paper proves exponential decay of correlations for two very important models of statistical physics: the random-cluster and the Potts model. This result is crucial for our understanding of the subcritical/disorder regime. Similar results were obtained in the eighties in the specific cases of Bernoulli percolation and the Ising model using BK-type correlation inequalities which are not available for general models. The technique using randomized algorithms introduced in this paper extends to a large variety of models. For instance, it allows to treat continuum percolation models such as Voronoi [40], Poisson-Boolean [50], Widom-Rowlinson [Dereudre-Houdebert], contact process, to cite but a few. It is also useful for many percolation processes obtained by taking the super-level lines of Gaussian fields; see for instance the case of the planar Bargmann-Fock process in [Muirhear-Vanneuville].

● Existence of phase transition for percolation using the Gaussian Free Field with S. Goswami, A. Raoufi, F. Severo and A. Yadin.

In this paper, we answer a conjecture raised by Benjamini and Schramm in their famous 1996 paper "Percolation beyond $\mathbb Z^d$: many questions and a few answers". Namely, we show that any Cayley graph with super-linear growth satisfies that $p_c<1$ for Bernoulli percolation (the existence of a non-trivial phase transition can be obtained for other models as well). The proof relies on a new connection between percolation and the Gaussian Free Field. This connection enables to relate the connectivity properties of percolation to geometric properties of the underlying non-necessarily transitive graph (such as isoperimetric dimension and Nash inequalities).

● Emergent Planarity in two-dimensional Ising Models with finite-range Interactions with M. Aizenman, V. Tassion and S. Warzel, to appear in

*Inventiones Mathematicae*.The known Pfaffian structure of the boundary spin correlations, and more generally order-disorder correlation functions, is given a new explanation through simple topological considerations within the model's random current representation. This perspective is then employed in the proof that the Pfaffian structure of boundary correlations emerges asymptotically at criticality in Ising models on $\mathbb Z^2$ with finite-range interactions. The analysis is enabled by new results on the stochastic geometry of the corresponding random currents. The proven statement establishes an aspect of universality, seen here in the emergence of fermionic structures in two dimensions beyond the solvable cases.

● Continuity of the phase transition for planar random-cluster and Potts models with $1\le q\le 4$ with V. Sidoravicius and V. Tassion, CMP (2017).

This paper develops a renormalization scheme for crossing probabilities in the critical random cluster model. Combined with the weak holomorphicity of parafermionic observables, one can prove that the phase transition of the random-cluster model with cluster-weight $q\in[1,4]$ is continuous. This answers half of a conjecture by Baxter about the behavior of the 2D Potts models. The property on crossing properties has many applications for the study of the critical model (polynomial decay of correlations, mixing properties, order of the phase transition, scaling relations). In particular, it enables one to prove that a number of mathematical definitions of a continuous phase transition are in fact equivalent. Since then, the renormalization techniques have been used in other models to prove a dichotomy result: either RSW-type estimates are valid, or there is exponential decay of correlations in the disordered phase (see e.g. [46]). The result was extended to isoradial graphs in [33].

● Discontinuity of the phase transition for the planar random-cluster and Potts models with $q>4$ with M. Gagnebin, M. Harel, I. Manolescu and V. Tassion.

This paper completes Baxter's conjecture on the behavior of the 2D Potts model by proving that the phase transition of the random-cluster model is discontinuous as soon as the cluster-weight $q$ is strictly larger than $4$. The techniques used in this paper are quite different from the one used in the companion paper (just above in the list). Here, we justified rigorously the Bethe Ansatz for the associated 6V model. Together with probabilistic techniques, this allowed us to compute the correlation length of the model explicitly at criticality, thus confirming a prediction of Baxter. The result was extended to isoradial graphs in [33].

● A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model with V. Tassion, CMP (2016).

This paper provides a completely new derivation, in three pages, of an important result in percolation theory dating from the 80', showing that probabilities of being connected decay exponentially fast in the subcritical regime. The proof extends to the Ising model. It covers the case of any transitive graph with arbitrary coupling constants. The strategy also implies that the susceptibility is infinite at criticality.

● Random currents and continuity of Ising model's spontaneous magnetization with M. Aizenman and V. Sidoravicius, CMP (2015).

This paper provides the first rigorous proof that the Ising model undergoes a continuous phase transition in 3D, and constitutes as such one of the only mathematical results on critical 3D systems. The result implies that the spin-spin correlations $\langle \sigma_0\sigma_x\rangle$ of the 3D Ising model at criticality decay slower than $C\|x\|^{-1}$ and faster than $c\|x\|^{-2}$. The proof relies on a geometric analysis of a graphical representation of the Ising model (such point of view can be used to prove a number of other results on the Ising model). The key of the proof is the introduction of a new percolation model, whose ergodic properties are related to the uniqueness of Gibbs measures at criticality; see the beautiful paper "Translation-Invariant Gibbs States of Ising model: General Setting" of Aran Raoufi for another application.

● The connective constant of the honeycomb lattice equals $\sqrt{2+\sqrt 2}$ with S. Smirnov, Ann. Math. (2012).

In 1980, the physicist Bernard Nienhuis predicted that the number $c_n$ of self-avoiding walks of length $n$ on the honeycomb lattice satisfies $c_n^{1/n}\rightarrow\sqrt{2+\sqrt 2}$ as $n$ tends to infinity. In this paper, we show this result by proving a weak form of discrete holomorphic for an observable of the model. The idea of harvesting a weak notion of discrete holomorphicity was then used in other papers dealing with self-avoiding walks, loop models, and (dependent) percolation models; see e.g. [3,6,17,19,46]. This paper represents one of the only available results in 2D (the value of the limit was conjectured by the physicist Bernard Nienhuis using the Coulomb Gas formalism in the early 80s). This result also constitutes a first step towards the proof of conformal invariance of the model on the hexagonal lattice; see [b,i]. Let us mention that understanding the typical geometric behavior of 2D self-avoiding walks is a beautiful mathematical challenge, see [12,26] for some (very partial) results and open questions.

● The self-dual point of the two-dimensional random-cluster model is critical for $q\ge1$ with V. Beffara, PTRF (2012).

In his celebrated paper of 1980, Kesten proved that the critical point of Bernoulli percolation on the square lattice $\mathbb Z^2$ is equal to $1/2$. The present paper provides the first rigorous computation of the critical point $p_c$ for an important dependent percolation model called the random-cluster model. This question goes back to the introduction of the model around 1969. The proof is based on a new Russo-Seymour-Welsh (RSW) result and sharp threshold theorems coming from the theory of boolean functions. The RSW theorem is one of the first RSW results for dependent percolation models. It was essential to the development of the theory at $p=p_c$, for instance in [30,41] (which present a more delicate description of the critical behavior). Related arguments (for instance proving the so-called sharpness of the phase transition) can be found in [24,34,38].

● The sharp threshold for bootstrap percolation in all dimensions with J. Balogh, B. Bollobás, and R. Morris, Trans. AMS (2012).

This paper shows the sharpness of the phase transition for a famous monotonic cellular automaton, called bootstrap percolation. The techniques (combining Probability Theory and Combinatorics) developed in the proof were later used to treat universality questions for a large class of monotonic automata, see e.g. [8,32,37,47,49].

● Connection probabilities and RSW-type bounds for the two-dimensional FK Ising model with C. Hongler, P. Nolin, CPAM (2011).

This paper contains the first example of a RSW-type theory for percolation models with dependency. The key is that the crossing probabilities are uniform in boundary conditions. The result was used in a variety of subsequent papers, for instance in the proof by Lubetzky and Sly that the Glauber dynamics of the critical Ising model mixes in polynomial time. Other applications can be found in [15,29]. In [23], the result was improved to cover all topological rectangles with non-degenerate extremal length. It was also instrumental in showing that the critical interfaces in the Ising and FK-Ising model on $\mathbb Z^2$ respectively converge to SLE(3) and SLE(16/3) (work [18] joint with Chelkak, Hongler, Kemppainen, and Smirnov). This latter result belongs to a short list of conformal invariance results for interfaces in 2D statistical physics.