### 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.

● Rotational invariance in critical planar lattice models with K.K. Kozlowski, D. Krachun, I. Manolescu, and M. Oulamara.

In this paper, we prove that the large scale properties of a number of two-dimensional lattice models are rotationally invariant. More precisely, we prove that the random-cluster model on the square lattice with cluster-weight $1\le q\le 4$ exhibits rotational invariance at large scales. This covers the case of Bernoulli percolation on the square lattice as an important example. We deduce from this result that the correlations of the Potts models with $q\in\{2,3,4\}$ colors and of the six-vertex height function with $\Delta\in[-1,-1/2]$ are rotationally invariant at large scales.

● Planar random-cluster model: scaling relations with I. Manolescu.

This result is the first derivation of generic scaling relations between critical exponents for a whole range of universality classes of lattice models. It deals with planar percolation models but we believe that the techniques can be extended to a number of other models. Scaling relations link different critical exponents describing the phase transition of systems from statistical physics. As such, they reduce the complexity of computing these exponents by explaining how they are all related to a few of them (in our case only two). In order to derive these scaling relations, we introduce a new link between these exponents and the rate of mixing of local observables in the system. The paper also paves the way to a number of new problems on the behaviour of the critical random cluster models, for instance the derivation of critical exponents, the construction of the near-critical scaling limit, the study of dynamical aspects.

● Marginal triviality of the scaling limits of critical 4D Ising and $\varphi_4^4$ models with M. Aizenman

The results of this paper address questions pertaining to two distinct research agendas: one aims at Constructive Field Theory and the other at the understanding of the critical behaviour in Statistical Mechanics. Our main result, from the perspective of euclidean field theory, is the triviality or gaussianity of Euclidean field theories obtained as the limit of critical (and near-critical) Ising and $\varphi_4^4$ models in dimension 4 (which corresponds to the space-time Minkowski space). This provides a complete answer to a program initiated in the eighties aiming at proving that there is no way of constructing non-trivial quantum field theories in four dimensions starting from the lattice $\varphi_4^4$ model. While an impossibility theorem was obtained independently in 82 by Aizenman and Frohlich in dimensions 5 and higher, this is the first unconditional result in four dimensions. What from the point of view of constructive field theory can look like a disapppointment is a positive result from the point of view of Statistical Mechanics, as it enables us to derive the scaling limit of critical 2n-point correlations in terms of the 2-point correlations.

● 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 for 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 [41], Poisson-Boolean [48], 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,

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) and

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.

The first 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. This result 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. [54]). The result was extended to isoradial graphs in [33] and general planar lattices [53]. The theory was recently extended to six-vertex high function [56], and was used for quantum spin chains in [45].

The second 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', stating that probabilities of being connected decay exponentially fast in the subcritical regime. The proof extends to the Ising model. It covers the case of quasi-transitive graphs 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. 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 [30,55] (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. [8,32,37,50].

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]

● Rotational invariance in critical planar lattice models with K.K. Kozlowski, D. Krachun, I. Manolescu, and M. Oulamara.

In this paper, we prove that the large scale properties of a number of two-dimensional lattice models are rotationally invariant. More precisely, we prove that the random-cluster model on the square lattice with cluster-weight $1\le q\le 4$ exhibits rotational invariance at large scales. This covers the case of Bernoulli percolation on the square lattice as an important example. We deduce from this result that the correlations of the Potts models with $q\in\{2,3,4\}$ colors and of the six-vertex height function with $\Delta\in[-1,-1/2]$ are rotationally invariant at large scales.

● Planar random-cluster model: scaling relations with I. Manolescu.

This result is the first derivation of generic scaling relations between critical exponents for a whole range of universality classes of lattice models. It deals with planar percolation models but we believe that the techniques can be extended to a number of other models. Scaling relations link different critical exponents describing the phase transition of systems from statistical physics. As such, they reduce the complexity of computing these exponents by explaining how they are all related to a few of them (in our case only two). In order to derive these scaling relations, we introduce a new link between these exponents and the rate of mixing of local observables in the system. The paper also paves the way to a number of new problems on the behaviour of the critical random cluster models, for instance the derivation of critical exponents, the construction of the near-critical scaling limit, the study of dynamical aspects.

● Marginal triviality of the scaling limits of critical 4D Ising and $\varphi_4^4$ models with M. Aizenman

*Ann. Math.*(to appear).The results of this paper address questions pertaining to two distinct research agendas: one aims at Constructive Field Theory and the other at the understanding of the critical behaviour in Statistical Mechanics. Our main result, from the perspective of euclidean field theory, is the triviality or gaussianity of Euclidean field theories obtained as the limit of critical (and near-critical) Ising and $\varphi_4^4$ models in dimension 4 (which corresponds to the space-time Minkowski space). This provides a complete answer to a program initiated in the eighties aiming at proving that there is no way of constructing non-trivial quantum field theories in four dimensions starting from the lattice $\varphi_4^4$ model. While an impossibility theorem was obtained independently in 82 by Aizenman and Frohlich in dimensions 5 and higher, this is the first unconditional result in four dimensions. What from the point of view of constructive field theory can look like a disapppointment is a positive result from the point of view of Statistical Mechanics, as it enables us to derive the scaling limit of critical 2n-point correlations in terms of the 2-point correlations.

● 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 for 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 [41], Poisson-Boolean [48], 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

*Duke Math. Journal*.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,

*Inventiones Mathematicae*(2019).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) and

*Annales de l'ENS*(to appear).The first 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. This result 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. [54]). The result was extended to isoradial graphs in [33] and general planar lattices [53]. The theory was recently extended to six-vertex high function [56], and was used for quantum spin chains in [45].

The second 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', stating that probabilities of being connected decay exponentially fast in the subcritical regime. The proof extends to the Ising model. It covers the case of quasi-transitive graphs 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. 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 [30,55] (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. [8,32,37,50].