### List of publications

**Universality for the random-cluster model on isoradial graphs.**

with Jhih-Huang Li and Ioan Manolescu, preprint.

**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.

**Macroscopic loops in the loop $O(n)$ model at Nienhuis' critical point.**

with Alexander Glazman, Ron Peled and Yinon Spinka, preprint.

**Abstract.** The loop $O(n)$
model is a model for a random collection of
non-intersecting loops on the hexagonal lattice, which
is believed to be in the same universality class as
the spin $O(n)$ model. It has been predicted by
Nienhuis that for $0\le n\le 2$ the loop $O(n)$ model
exhibits a phase transition at a critical parameter
$x_c(n)=1/\sqrt{2+\sqrt{2-n}}$. For $0 < n\le 2$,
the transition line has been further conjectured to
separate a regime with short loops when $x <
x_c(n)$ from a regime with macroscopic loops when
$x\ge x_c(n)$. In this paper, we prove that for $n\in
[1,2]$ and $x=x_c(n)$ the loop $O(n)$ model exhibits
macroscopic loops. This is the first instance in which
a loop $O(n)$ model with $n\neq 1$ is shown to exhibit
such behaviour. A main tool in the proof is a new
positive association (FKG) property shown to hold when
$n \ge 1$ and $0 < x\le 1/\sqrt{n}$. This property
implies, using techniques recently developed for the
random-cluster model, the following dichotomy: either
long loops are exponentially unlikely or the origin is
surrounded by loops at any scale (box-crossing
property). We develop a ``domain gluing'' technique
which allows us to employ Smirnov's parafermionic
observable to rule out the first alternative when
$x=x_c(n)$ and $n\in[1,2]$.

**Internal Diffusion-Limited aggregation with uniform starting points.**

with Itai Benjamini, Gady Kozma and Cyrille Lucas, preprint.

**Abstract.** We study internal
diffusion-limited aggregation with random starting
points on $\mathbb Z^d$. In this model, each new
particle starts from a vertex chosen uniformly at random
on the existing aggregate. We prove that the limiting
shape of the aggregate is a Euclidean ball.

**A note on Schramm's locality conjecture for random-cluster models.**

with Vincent Tassion, preprint.

**Abstract.** In this note, we
discuss a generalization of Schramm's locality
conjecture to the case of random-cluster models. We give
some partial (modest) answers, and present several
related open questions. Our main result is to show that
the critical inverse temperature of the Potts model on
$\mathbb Z^r\times(\mathbb Z/2n\mathbb Z)^{d-r}$ (with
$r\ge3$) converges to the critical inverse temperature
of the model on $\mathbb Z^d$ as $n$ tends to infinity.
Our proof relies on the infrared bound and, contrary to
the equivalent (harder) statement for Bernoulli
percolation, does not involve renormalization arguments.

**Lectures on the Ising and Potts models on the hypercubic lattice.**

preprint.

**Abstract.** Phase transitions
are a central theme of statistical mechanics, and of
probability more generally. Lattice spin models
represent a general paradigm for phase transitions in
finite dimensions, describing ferromagnets and even
some fluids (lattice gases). It has been understood
since the 1980s that random geometric representations,
such as the random walk and random current
representations, are powerful tools to understand spin
models. In addition to techniques intrinsic to spin
models, such representations provide access to rich
ideas from percolation theory. In recent years, for
two-dimensional spin models, these ideas have been
further combined with ideas from discrete complex
analysis. Spectacular results obtained through these
connections include the proofs that interfaces of the
two-dimensional Ising model have conformally invariant
scaling limits given by SLE curves, that the
connective constant of the self-avoiding walk on the
hexagonal lattice is given by $\sqrt{2+\sqrt 2}$. In
higher dimensions, the understanding also progresses
with the proof that the phase transition of Potts
models is sharp, and that the magnetization of the
three-dimensional Ising model vanishes at the critical
point.

**Comment.**These lecture notes describe the content of a class given at the PIMS-CRM probability summer school on the behavior of lattice spin models near their critical point.

**Exponential decay of connection probabilities for subcritical Voronoi percolation in $\mathbb R^d$.**

with Aran Raoufi and Vincent Tassion, preprint.

**Abstract.** We prove that for
Voronoi percolation on $\mathbb R^d$, there exists
$p_c\in[0,1]$ such that

- for $p$ smaller than $p_c$, there exists $c_p>0$
such that $\mathbb P_p[0\text{ connected to distance
}n]\le \exp(-c_pn)$,

- there exists $c>0$ such that for $p>p_c$,
$\mathbb P_p[0\text{ connected to }\infty]\ge
c(p-p_c)$.

This result offers a new way of showing that
$p_c=1/2$, and the first proof of mean-field lower
bound for the density of the infinite cluster, even in
dimension 2. This paper belongs to a series of papers
using the theory of algorithms to prove sharpness of
the phase transition.

**Sharp phase transition for the random-cluster and Potts models via decision trees.**

with Aran Raoufi and Vincent Tassion, preprint.

**Abstract.** We prove an
inequality on decision trees on monotonic measures
which generalizes the OSSS inequality on product
spaces. As an application, we use this inequality to
prove a number of new results on lattice spin models
and their random-cluster representations. More
precisely, we prove that

- For the Potts model on transitive graphs,
correlations decay exponentially fast below
criticality.

- For the random-cluster model with cluster weight
$q\ge1$ on transitive graphs, correlations decay
exponentially fast in the subcritical regime and the
cluster-density satisfies the mean-field lower bound
in the supercritical regime.

- For the random-cluster models with cluster weight
$q\ge1$ on planar quasi-transitive graphs $\mathbb G$,

$\frac{p_c(\mathbb G)p_c(\mathbb G^*)}{(1-p_c(\mathbb
G))(1-p_c(\mathbb G^*))}~=~q.$

As a special case, we obtain the value of the critical
point for the square, triangular and hexagonal
lattices (this provides a short proof of the result of
Beffara and Duminil-Copin. These results have many
applications for the understanding of the subcritical
(respectively disordered) phase of all these models.
The techniques developed in this paper have potential
to be extended to a wide class of models including the
Ashkin-Teller model, continuum percolation models such
as Voronoi percolation and Boolean percolation,
super-level sets of massive Gaussian Free Field, and
random-cluster and Potts model with infinite range
interactions.

**Discontinuity of the phase transition for the planar random-cluster and Potts models with $q>4$.**

with Maxime Gagnebin, Matan Harel, Ioan Manolescu and Vincent Tassion, preprint.

© Vincent Beffara

**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

1. Existence of multiple infinite-volume measures for
the critical Potts and random-cluster models,

2. Ordering for the measures with monochromatic (resp.
wired) boundary conditions for the critical Potts
model (resp. random-cluster model), and

3. 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 Maxime Gagnebin, Matan Harel, Ioan Manolescu and Vincent Tassion, preprint.

**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 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.

**Universality of two-dimensional critical cellular automata.**

with B. Bollobás, R. Morris and P. Smith, preprint.

**Abstract.** We study the class
of monotone, two-state, deterministic cellular
automata, in which sites are activated (or `infected')
by certain congurations of nearby infected sites.
These models have close connections to statistical
physics, and several specific examples have been
extensively studied in recent years by both
mathematicians and physicists. This general setting
was first studied only recently, however, by
Bollobas, Smith and Uzzell, who showed that the
family of all such `bootstrap percolation' models on
$\mathbb Z^2$ can be naturally partitioned into three
classes, which they termed subcritical, critical and
supercritical.

In this paper we determine the order of the threshold
for percolation (complete occupation) for every
critical bootstrap percolation model in two
dimensions. This `universality' theorem includes as
special cases results of Aizenman and Lebowitz,
Gravner and Grieath, Mountford, and van Enter and
Hulshof, signicantly strengthens bounds of Bollobas,
Smith and Uzzell, and complements recent work of
Balister, Bollobas, Przykucki and Smith on
subcritical models.

**Finite volume Bootstrap Percolation with balanced threshold rules on $\mathbb{Z}^2$.**

with A. Holroyd, preprint.

**Abstract.** We prove that there
exists a sharp metastability transition for
two-dimensional ﬁnite bootstrap percolation with
threshold growth rules (and convex neighborhood). This
result extends theorems obtained by Holroyd for the
simple two-dimensional Bootstrap Percolation. The
method emphasizes physical phenomena involved in the
growth of the dynamic and oﬀers an expandable frame
that may be used to prove sharp metastability
transition results for a wide class of cellular
automata, see for an introduction to this class of
models. This article represents a further step towards
an understanding of universality in two dimensional
bootstrap models.

**Higher order corrections for anisotropic bootstrap percolation.**

with A. C. D. van Enter and T. Hulshof, to appear in Probability Theory and Related Fields.

© Tim Hulshof

**Abstract.** We study the
critical probability for the metastable phase
transition of the two-dimensional anisotropic
bootstrap percolation model with $(1,2)$-neighbourhood
and threshold $r = 3$. The first order asymptotics for
the critical probability were recently determined by
the first and second authors. Here we determine the
following sharp second and third order asymptotics: \[

p_c\big( [L]^2,\mathcal{N}_{(1,2)},3 \big) \; = \;
\frac{(\log \log L)^2}{12\log L} \, - \, \frac{\log
\log L \, \log \log \log L}{ 3\log L}

+ \frac{\left(\log \frac{9}{2} + 1 \pm o(1)
\right)\log \log L}{6\log L}. \] We note that the
second and third order terms are so large that the
first order asymptotics fail to approximate $p_c$ even
for lattices of size well beyond $10^{10^{1000}}$.

**The sharp threshold for the Duarte model.**

with B. Bollobás, R. Morris and P. Smith, to appear in Annals of Probability.

**Abstract.** The class of
critical bootstrap percolation models in two
dimensions was recently introduced by Bollobás, Smith
and Uzzell, and the critical threshold for percolation
was determined up to a constant factor for all such
models by the authors of this paper. Here we develop
and refine the techniques introduced in that paper in
order to determine a sharp threshold for the Duarte
model. This resolves a question of Mountford from
1995, and is the first result of its type for a model
with drift.

**The box-crossing property for critical two-dimensional oriented percolation.**

with V. Tassion and A. Teixeira, to appear in Probability Theory and Related Fields.

**Abstract.** We consider critical
oriented Bernoulli percolation on the square lattice
$\mathbb{Z}^2$. We prove a Russo-Seymour-Welsh type result
which allows us to derive several new results concerning
the critical behavior: - We establish that the probability
that the origin is connected to distance $n$ decays
polynomially fast in $n$. - We prove that the critical
cluster of the origin conditioned to survive to distance
$n$ has a typical width $w_n$ satisfying $\epsilon n^{2/5}
< w_n < n^{1-\epsilon}$ for some $\epsilon > 0$.
The sub-linear polynomial fluctuations contrast with the
supercritical regime where $w_n$ is known to behave
linearly in $n$. It is also different from the critical
picture obtained for non-oriented Bernoulli percolation,
in which the scaling limit is non-degenerate in both
directions. All our results extend to the graphical
representation of the one-dimensional contact process.

© Vincent Tassion

**Minimal growth harmonic functions on lamplighter groups.**

with I. Benjamini, G. Kozma and A. Yadin, to appear in New York J Math.

**Abstract.** We study the minimal
possible growth of harmonic functions on lamplighters. We
find that $(\mathbb Z/2)\wr \mathbb Z$ has no sublinear
harmonic functions, $(\mathbb Z/2)\wr \mathbb Z^2$ has no
sublogarithmic harmonic functions, and neither has the
repeated wreath product $(\dotsb(\mathbb Z/2\wr\mathbb
Z^2)\wr\mathbb Z^2)\wr\dotsb\wr\mathbb Z^2$. These results
have implications on attempts to quantify the
Derriennic-Kaimanovich-Vershik theorem.

**Brochette percolation.**

with M. Hilario, G. Kozma and V. Sidoravicius, to appear in Israel Journal of Mathematics.

**Abstract.** We study bond
percolation on the square lattice with one-dimensional
inhomogeneities. Inhomogeneities are introduced in the
following way: A vertical column on the square lattice is
the set of vertical edges that project to the same vertex
on $\mathbb{Z}$. Select vertical columns at random
independently with a given positive probability. Keep
(respectively remove) vertical edges in the selected
columns, with probability $p$, (respectively $1-p$). All
horizontal edges and vertical edges lying in unselected
columns are kept (respectively removed) with probability
$q$, (respectively $1-q$). We show that, if $p >
p_c(\mathbb{Z}^2)$ (the critical point for homogeneous
Bernoulli bond percolation) then $q$ can be taken strictly
smaller then $p_c(\mathbb{Z}^2)$ in such a way that the
probability that the origin percolates is still positive.

**A new computation of the critical point for the planar random-cluster model with $q\ge1$.**

with A. Raoufi and V. Tassion, to appear in Annales de l'IHP.

**Abstract.** We present a new
computation of the critical value of the random-cluster
model with cluster weight $q\ge 1$ on $\mathbb Z^2$. This
provides an alternative approach to the result obtained by
Beffara and Duminil-Copin. We believe that this approach
has several advantages. First, most of the proof can
easily be extended to other planar graphs with sufficient
symmetries. Furthermore, it invokes RSW-type arguments
which are not based on self-duality. And finally, it
contains a new way of applying sharp threshold results
which avoid the use of symmetric events and periodic
boundary conditions. Some of the new methods presented in
this paper have a larger scope than the planar
random-cluster model, and may be useful to investigate
sharp threshold phenomena for more general dependent
percolation processes, in any dimension.

2017

**Exponential decay of loop lengths in the loop $ O (n) $ model with large $ n$.**

with R. Peled, W. Samotij, Y. Spinka, Communications in Mathematical Physics, 349(3), 777- 817, 2017.

**Abstract.** The loop $O(n)$
model is a model for a random collection of
non-intersecting loops on the hexagonal lattice, which
is believed to be in the same universality class as
the spin $O(n)$ model. It has been conjectured that
both the spin and the loop $O(n)$ models exhibit
exponential decay of correlations when $n>2$. We
verify this for the loop $O(n)$ model with large
parameter $n$, showing that long loops are
exponentially unlikely to occur, uniformly in the edge
weight $x$. Our proof provides further detail on the
structure of typical configurations in this regime.
Putting appropriate boundary conditions, when $nx^6$
is sufficiently small, the model is in a dilute,
disordered phase in which each vertex is unlikely to
be surrounded by any loops, whereas when $nx^6$ is
sufficiently large, the model is in a dense, ordered
phase which is a small perturbation of one of the
three ground states.

**Continuity of the phase transition for planar random-cluster and Potts models with $1\le q\le 4$.**

with V. Sidoravicius and V. Tassion, Communications in Mathematical Physics, 349(1), 47-107, 2017.

**Abstract.** This article studies
the planar Potts model and its random-cluster
representation. We show that the phase transition of
the nearest-neighbor ferromagnetic $q$-state Potts
model on $\mathbb Z^2$ is continuous for
$q\in\{2,3,4\}$, in the sense that there exists a
unique Gibbs state, or equivalently that there is no
ordering for the critical Gibbs states with
monochromatic boundary conditions. The proof uses the
random-cluster model with cluster-weight $q\ge1$ (note
that $q$ is not necessarily an integer) and is based
on two ingredients: The fact that two-point function
for the free state decays sub-exponentially fast for
cluster-weights $1\le q\le 4$, which is derived
studying parafermionic observables on a discrete
Riemann surface. And a new result proving the
equivalence of several properties of critical
random-cluster models: - the absence of
infinite-cluster for wired boundary conditions,

- the uniqueness of infinite-volume measures,

- the sub-exponential decay of the two-point function
for free boundary conditions,

- a Russo-Seymour-Welsh type result on crossing
probabilities in rectangles with {\em arbitrary
boundary conditions}.

The result leads to a number of consequences
concerning the scaling limit of the random-cluster
model with $q\in[1,4]$. It shows that the family of
interfaces (for instance for Dobrushin boundary
conditions) are tight when taking the scaling limit
and that any sub-sequential limit can be parametrized
by a Loewner chain. We also study the effect of
boundary conditions on these sub-sequential limits.
Let us mention that the result should be instrumental
in the study of critical exponents as well.

2016

**Conformal invariance of crossing probabilities for the Ising model with free boundary conditions.**

with S. Benoist and C. Hongler, Annales de l'Institut Henri Poincaré, 52(4), 1784-1798, 2016.

**Abstract.** We prove that crossing
probabilities for the critical planar Ising model with
free boundary conditions are conformally invariant in the
scaling limit. We do so by establishing the convergence of
certain exploration processes towards
SLE$(3,\frac{-3}2,\frac{-3}2)$. We also derive results on
the exploration tree introduced by Sheffield.

**A new proof of the sharpness of the phase transition for Bernoulli percolation on $\mathbb Z^d$.**

with V. Tassion, L'Enseignement Mathématique, 62(1/2), 199-206, 2016.

**Abstract.** We provide a new proof
of the sharpness of the phase transition for
nearest-neighbour Bernoulli percolation. More precisely,
we show that for $p < p_c$, the probability that the
origin is connected by an open path to distance $n$ decays
exponentially fast in $n$, and for $p>p_c$, the
probability that the origin belongs to an infinite cluster
satisfies the mean-field lower bound $\theta(p)\ge
\tfrac{p-p_c}{p(1-p_c)}$. This note presents the argument
of "A new proof of sharpness of the phase transition for
Bernoulli percolation and the Ising model", which is valid
for long-range Bernoulli percolation (and for the Ising
model) on arbitrary transitive graphs in the simpler
framework of nearest-neighbour Bernoulli percolation on
$\mathbb Z^d$.

**A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model.**

with V. Tassion, Communications in Mathematical Physics, 343(2), 725-745, 2016.

**Abstract.** We provide a new proof
of the sharpness of the phase transition for Bernoulli
percolation and the Ising model. The proof applies to
infinite range models on arbitrary locally finite
transitive infinite graphs. For Bernoulli percolation, we
prove finiteness of the susceptibility in the subcritical
regime $\beta<\beta_c$, and the mean-field lower bound
$\mathbb P_\beta[0\longleftrightarrow\infty]\ge
(\beta-\beta_c)/\beta$ for $\beta>\beta_c$. For
finite-range models, we also prove that for any
$\beta<\beta_c$, the probability of an open path from
the origin to distance $n$ decays exponentially fast in
$n$. For the Ising model, we prove finiteness of the
susceptibility for $\beta<\beta_c$, and the mean-field
lower bound $\langle \sigma_0\rangle_\beta^+\ge
\sqrt{(\beta^2-\beta_c^2)/\beta^2}$ for
$\beta>\beta_c$. For finite-range models, we also prove
that the two-point correlations functions decay
exponentially fast in the distance for $\beta<\beta_c$.

**On the probability that self-avoiding walk ends at a given point.**

with A. Glazman, A. Hammond, and I. Manolescu, Annals of probability, 44(2), 955-983, 2016.

**Abstract.** We prove two results on
the delocalization of the endpoint of a uniform
self-avoiding walk on $\mathbb Z^d$ for $d\ge2$. 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 $\|x\|=1$, this
probability decreases faster than $n^{-1/4+\varepsilon}$
for any $\varepsilon>0$. This provides a bound on the
probability that a self-avoiding walk is a polygon.

**A quantitative Burton-Keane estimate under strong FKG condition.**

with D. Ioffe and Y. Velenik, Annals of Probability, 44(5), 3335-3356, 2016.

**Abstract.** We consider
translationally-invariant percolation models on
$\mathbb Z^d$ satisfying the finite energy and the FKG
properties. We provide explicit upper bounds on the
probability of having two distinct clusters going from
the endpoints of an edge to distance $n$ (this
corresponds to a finite size version of the celebrated
Burton-Keane argument proving uniqueness of the
infinite-cluster). The proof is based on the
generalization of a reverse Poincaré inequality proved
by Sen and Chatterjee. As a consequence, we obtain
upper bounds on the probability of the so-called
four-arm event for planar random-cluster models with
cluster-weight $q\geq1$.

**The phase transitions of the planar random-cluster and Potts models with $q\ge1$ are sharp.**

with I.Manolescu, Probability Theory and Related Fields, 164(3), 865-892, 2016.

**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 $p_c$ 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. Due to its
nature, this strategy could be useful in studying
other planar models satisfying the FKG lattice
condition and some additional differential
inequalities.

**Crossing probabilities in topological rectangles for the critical planar FK-Ising model.**

with D. Chelkak and C. Hongler, Electronic Journal of Probability, 21(1), 1-28, 2016.

**Abstract.** We consider the FK-Ising
model in two dimensions at criticality. We obtain RSW-type
crossing probabilities bounds in arbitrary topological
rectangles, uniform with respect to the boundary
conditions, generalizing existing results. Our result
relies on new discrete complex analysis techniques,
introduced in a previous paper by Chelkak. We detail some
applications, in particular the computation of so-called
universal exponents and crossing bounds for the classical
Ising model.

**Absence of infinite cluster for critical Bernoulli percolation on slabs.**

with V. Sidoravicius and V. Tassion, Communications in Pure and Applied Mathematics, 69(7), 1397-1411, 2016.

**Abstract.** We prove that for
Bernoulli percolation on a graph $\mathbb
Z^2\times\{0,\dots,k\}$, $k\ge0$, there is no infinite
cluster at criticality, almost surely. The proof extends
to finite range Bernoulli percolation models on $\mathbb
Z^2$ which are invariant under $\pi/2$-rotation and
reflection.

2015

**Disorder, entropy and harmonic functions.**

with I. Benjamini, G. Kozma and A. Yadin, Annals of Probability, 43(5), 2332-2373, 2015.

**Abstract.** We study harmonic
functions on random environments with particular
emphasis on the case of the infinite cluster of
supercritical percolation on $\mathbb Z^d$. We prove
that the vector space of harmonic functions growing at
most linearly is $d+1$-dimensional almost surely. In
particular, there are no non-constant sublinear
harmonic functions (thus implying the uniqueness of
the corrector). The main ingredient of the proof is
given by a quantitative, annealed version of the
Kaimanovich-Vershik entropy argument. This also
provides bounds on the derivative of the heat kernel,
simplifying and generalizing existing results. Even
reversibility is not necessary. We also mention
several open problems and conjectures on the behavior
of harmonic functions on stationary random graphs.

**Random currents and continuity of Ising model's spontaneous magnetization.**

with M. Aizenman and V. Sidoravicius, Communications in Mathematical Physics, 334, 719-742, 2015.

**Abstract.** The spontaneous
magnetization is proved to vanish continuously at the
critical temperature for a class of ferromagnetic
Ising spin systems which includes the nearest neighbor
ferromagnetic Ising spin model on $\mathbb Z^d$ in
$d=3$ dimensions. The analysis applies also to higher
dimensions, for which the result is already known, and
to systems with interactions of power law decay. The
proof employs in an essential way an extension of
Ising model's *random current representation* to
the model's infinite volume limit. Using it, we relate
the continuity of the magnetization to the vanishing
of the free boundary condition Gibbs state's Long
Range Order parameter. For reflection positive models
the resulting criterion for continuity may be
established through the infrared bound for all but the
borderline case, of the one dimensional model with
$1/r^2$ interaction, for which the spontaneous
magnetization is known to be discontinuous at $T_c$.

**On the critical parameters of the $q\ge4$ random-cluster model on isoradial graphs.**

with V. Beffara and S. Smirnov, Journal Physics A: Mathematical and Theoretical, 48(48), 484003, 2015.

**Abstract.** The critical surface
for random-cluster model with cluster-weight $q\ge 4$
on isoradial graphs is identified using parafermionic
observables. Correlations are also shown to decay
exponentially fast in the subcritical regime. While
this result is restricted to random-cluster models
with $q\ge 4$, it extends the recent theorem of [BD12]
to a large class of planar graphs. In particular, the
anisotropic random-cluster model on the square lattice
are shown to be critical if
$\frac{p_vp_h}{(1-p_v)(1-p_h)}=q$, where $p_v$ and
$p_h$ denote the horizontal and vertical edge-weights
respectively. We also provide consequences for Potts
models.

**Comment.** This article is part of the special
issue **Exactly solved models and beyond** in
honour of R. Baxter's 75th birthday.

2014

**Convergence of Ising interfaces to Schramm's SLE curves.**

with D. Chelkak, C. Hongler, A. Kemppainen and S. Smirnov, Comptes Rendus Mathematique, 352(2), 157-161, 2014.

**Abstract.** We show how to combine
our earlier results to deduce strong convergence of the
interfaces in the planar critical Ising model and its
random-cluster representation to Schramm’s SLE curves with
parameter $\kappa=3$ and $\kappa=16/3$ respectively.

**The critical fugacity for surface adsorption of self-avoiding walks on the honeycomb lattice is $1+\sqrt 2$.**

with N. Beaton, M. Bousquet-Mélou, J. De Gier and A. Guttmann, Communications in Mathematical Physics, 326(3), 727-754, 2014.

**Abstract.** In 2010, the growth
constant of self-avoiding walks on the hexagonal
(a.k.a honeycomb) lattice was proved to be equal to
$\sqrt{2+\sqrt 2}$: A key identity used in that proof
was later generalised by Smirnov so as to apply to a
general $O(n)$ loop model with $n\in[-2,2]$ (the case
$n = 0$ corresponding to self-avoiding walks). We
modify this model by restricting to a half-plane and
introducing a fugacity associated with boundary sites
(also called surface sites) and obtain a
generalisation of Smirnov’s identity. The value of the
critical surface fugacity was conjectured by Batchelor
and Yung in 1995. This value also plays a crucial role
in our identity, which thus provides an independent
prediction for it. For the case $n = 0$, corresponding
to self-avoiding walks interacting with a surface, we
prove the conjectured value of the critical surface
fugacity. A critical part of this proof involves
demonstrating that the generating function of
self-avoiding bridges of height $T$, taken at its
critical point, tends to 0 as $T$ increases, as
predicted from SLE theory.

**Seven-dimensional forest fires.**

with D. Ahlberg, G. Kozma and V. Sidoravicius, Annales de l'Institut Henri Poincaré, 50(2), 315-326, 2014.

**Abstract.** We show that in high
dimensional Bernoulli percolation, removing from a thin
infinite cluster a much thinner infinite cluster leaves an
infinite component. This observation has implications for
the van den Berg-Brouwer forest fire process, also known
as self-destructive percolation, for dimension high
enough.

**The near-critical planar FK-Ising model.**

with C. Garban and G. Pete, Communications in Mathematical Physics, 326, 1-35, 2014.

**Abstract.** We study the
near-critical FK-Ising model. First, a determination of
the correlation length defined via crossing probabilities
is provided. Second, a phenomenon about the near-critical
behavior of FK-Ising is highlighted, which is completely
missing from the case of standard percolation: in any
monotone coupling of FK configurations (e.g., in the one
introduced by Grimmett), as one raises $p$ near $p_c$, the
new edges arrive in a self-organized way, so that the
correlation length is not governed anymore by the number
of pivotal edges at criticality.

**Supercritical self-avoiding walks are space-filling.**

with G. Kozma and A. Yadin, Annales de l'Institut Henri Poincaré, 50(2), 315-326, 2014.

**Abstract.** We consider random
self-avoiding walks between two points on the boundary of
a finite subdomain of $\mathbb Z^d$ (the probability of a
self-avoiding trajectory gamma is proportional to
$\mu^{-\ell(\gamma)}$). We show that the random trajectory
becomes space-filling in the scaling limit when the
parameter mu is supercritical.

**On the Gibbs states of the noncritical Potts model on $\mathbb Z^2$.**

with L. Coquille, D. Ioffe and Y. Velenik, Probability Theory and Related Fields, 158(1-2), 477-512, 2014.

**Abstract.** We prove that all
Gibbs states of the q-state nearest neighbor Potts
model on $\mathbb Z^2$ below the critical temperature
are convex combinations of the $q$ pure phases; in
particular, they are all translation invariant. To
achieve this goal, we consider such models in large
finite boxes with arbitrary boundary condition, and
prove that the center of the box lies deeply inside a
pure phase with high probability. Our estimate of the
finite-volume error term is of essentially optimal
order, which stems from the Brownian scaling of
fluctuating interfaces. The results hold at any
supercritical value of the inverse temperature $\beta
\ge\beta_c$.

© Vincent Beffara

2013

**The critical temperature for the Ising model on planar doubly periodic graphs.**

with D. Cimasoni, Electronic Journal in Probability, 18(44), 1-18, 2013.

**Abstract.** We provide a simple
characterization of the critical temperature for the Ising
model on an arbitrary planar doubly periodic weighted
graph. More precisely, the critical inverse temperature
$\beta$ for a graph $G$ with coupling constants
$(J_e)_{e\in E(G)}$ is obtained as the unique solution of
a linear equation in the variables $(\tanh(\beta
J_e))_{e\in E(G)}$. This is achieved by studying the
high-temperature expansion of the model using Kac-Ward
matrices.

**Limit of the Wulff Crystal when approaching criticality for site percolation on the triangular lattice.**

Electronic Communications in Probability, 18(93), 1–9, 2013.

**Abstract.** When a model is
conformally invariant (hence rotationally invariant) at
criticality, theWulff crystal is expected to become
isotropic as the parameter tends to the critical one.
Nevertheless, known proofs of conformal invariance give
very little information on the isotropy of off critical
regimes. The aim of this small note is to derive the
convergence of the Wulff crystal to a disk for site
percolation on the triangular lattice. The main ingredient
of the proof is the result of Garban, Pete and Schramm
[GPS] on the rotational invariance of the near-critical
scaling limit.

**Self-avoiding walk is sub-ballistic.**

with A. Hammond, Communications in Mathematical Physics, 324(2), 401-423, 2013.

**Abstract.** We prove that
self-avoiding walk on $\mathbb Z^d$ is sub-ballistic in
any dimension $d\ge2$.

**Containing Internal Diffusion Limited Aggregation.**

with C. Lucas, A. Yadin and A. Yehudayoff, Electronic Communications in Probability, 18(50), 1-8, 2013.

**Abstract.** Internal Diffusion
Limited Aggregation (IDLA) is a model that describes
the growth of a random aggregate of particles from the
inside out. Shellef proved that IDLA processes on
supercritical percolation clusters of integer-lattices
fill Euclidean balls, with high probability. In this
article, we complete the picture and prove a
limit-shape theorem for IDLA on such percolation
clusters, by providing the corresponding upper bound.
The technique to prove upper bounds is new and robust:
it only requires the existence of a good lower bound.
Specifically, this way of proving upper bounds on
IDLA clusters is more suitable for random environments
than previous ways, since it does not harness harmonic
measure estimates.

© Cyrille Lucas

**Sharp metastability threshold for an anisotropic bootstrap percolation model.**

with A. C. D. van Enter, Annals of Probability, 41(3A), 1218-1242, 2013.

**Abstract.** Bootstrap
percolation models have been extensively studied
during the two past decades. In this article, we study
an anisotropic bootstrap percolation model. We prove
that it exhibits a sharp metastability threshold. This
is the first mathematical proof of a sharp threshold
for an anisotropic bootstrap percolation model.

**Erratum.** Although Theorem 1.1 as stated in our
paper is correct, the generalisation announced at the
end of page 1219 and the beginning of page 1220 to the
situation where the site $(m,n)$ gets occupied if
$k+1$ sites among the $2k+2$ sites $(m+1,n)$,
$(m,n-1)$ and $(m-k,n),\dots, (m-1,n),
(m+1,n),\dots,(m+k,n)$ are occupied, is incorrect as
stated for $k\ge2$. It should read that (in
probability) $\frac{1}{p}(\log \frac{1}{p})^2 \log T$
tends to $\frac{(k-1)^2}{4(k+1)}$. In other words, the
constant $\frac{1}{4(k+1)}$ given in our paper on the
right-hand side of the unnumbered equation on page
1220 should be replaced by $\frac{(k-1)^2}{4(k+1)}$.
We thank Rob Morris for pointing this out to us.

Also, the definition of weakly connected in page 1226
should be modified as follows. Two occupied points
$x,y\in\mathbb Z^2$ are weakly connected if $x\in
y+\mathcal N$ or there exists $z\in \mathbb Z^2$ such
that $x,y\in z+\mathcal N$. We thank Tim Hulshof for
pointing this out to us.

2012

**Divergence of the correlation length for critical planar FK percolation with $1\le q\le 4$ via parafermionic observables.**

Journal Physics A: Mathematical and Theoretical, 45 494013, 26 pages, 2012.

**Abstract.** Parafermionic
observables were introduced by Smirnov for planar FK
percolation in order to study the critical phase $(p;
q) = (p_c(q); q)$. This article gathers several known
properties of these observables. Some of these
properties are used to prove the divergence of the
correlation length when approaching the critical point
for FK percolation when $1\le q\le 4$. A crucial step
is to consider FK percolation on the universal cover
of the punctured plane. We also mention several
conjectures on FK percolation with arbitrary
cluster-weight $q \ge 0$.

**Comment.** This article is part of 'Lattice
models and integrability', a special issue of Journal
of Physics A: Mathematical and Theoretical in honour
of F Y Wu's 80th birthday.

**The self-dual point of the two-dimensional random-cluster model is critical for $q\ge1$.**

with V. Beffara, Probability Theory and Related Fields, 153(3), 511-542, 2012.

**Abstract.** We prove a
long-standing conjecture on random-cluster models,
namely that the critical point for such models with
parameter $q\ge1$ on the square lattice is equal to
the self-dual point $p_{\rm
sd}(q)=\sqrt{q}/(1+\sqrt{q})$. This gives a rigorous
proof that the critical temperature of the q-state
Potts model is equal to $\log(1+\sqrt q)$ for all
$q\ge 1$. We further prove that the transition is
sharp, meaning that there is exponential decay of
correlations in the sub-critical phase. The techniques
of this paper are rigorous and valid for all $q\ge1$,
in contrast to earlier methods valid only for certain
given $q$. The proof extends to the triangular and the
hexagonal lattices as well.

**The connective constant of the honeycomb lattice equals $\sqrt{2+\sqrt 2}$.**

with S. Smirnov, Annals of Mathematics (2), 175(3), 1653-1665, 2012.

**Abstract.** We provide the
first mathematical proof that the connective constant
of the hexagonal lattice is equal to $\sqrt{2+\sqrt
2}$. This value has been derived non rigorously by B.
Nienhuis in 1982, using Coulomb gas approach from
theoretical physics. Our proof uses a parafermionic
observable for the self avoiding walk, which
satisfies a half of the discrete Cauchy-Riemann
relations. Establishing the other half of the
relations (which conjecturally holds in the scaling
limit) would also imply convergence of the
self-avoiding walk to SLE(8/3).

© Vincent Beffara

**The sharp threshold for bootstrap percolation in all dimensions.**

with J. Balogh, B. Bollobás, and R. Morris, Transaction of the American Mathematical Society, 364, 2667-2701, 2012.

**Abstract.** In $r$-neighbour
bootstrap percolation on a graph $G$, a (typically
random) set $A$ of initially 'infected' vertices
spreads by infecting (at each time step) vertices with
at least $r$ already-infected neighbours. This process
may be viewed as a monotone version of the Glauber
dynamics of the Ising model, and has been extensively
studied on the $d$-dimensional grid $[n]^d$. The
elements of the set $A$ are usually chosen
independently, with some density $p$, and the main
question is to determine $p_c([n]^d,r)$, the density
at which percolation (infection of the entire vertex
set) becomes likely.

In this paper we prove, for every pair $d \ge r \ge
2$, that there is a constant $L(d,r)$ such that
$p_c([n]^d,r) = [(L(d,r) + o(1)) / log_(r-1)
(n)]^{d-r+1}$ as $n \to \infty$, where $log_r$ denotes
an $r$-times iterated logarithm. We thus prove the
existence of a sharp threshold for percolation in any
(fixed) number of dimensions. Moreover, we determine
$L(d,r)$ for every pair $(d,r)$.

**Smirnov's fermionic observable away from criticality.**

with V. Beffara, Annals of Probability, 40(6), 2667-2689, 2012.

**Abstract.** In a recent and
celebrated article, Smirnov deﬁnes an observable for the
self-dual random-cluster model with cluster weight $q = 2$
on the square lattice $\mathbb Z^2$ , and uses it to
obtain conformal invariance in the scaling limit. We study
this observable away from the self-dual point. From this,
we obtain a new derivation of the fact that the self-dual
and critical points coincide, which implies that the
critical inverse temperature of the Ising model equals
$\tfrac12 \log(1 + \sqrt 2)$. Moreover, we relate the
correlation length of the model to the large deviation
behavior of a certain massive random walk (thus conﬁrming
an observation by Messikh), which allows us to compute it
explicitly.

2011

**Connection probabilities and RSW-type bounds for the two-dimensional FK Ising model.**

with C. Hongler, P. Nolin, Communications in Pure and Applied Mathematics, 64(9), 1165-1198, 2011.

**Abstract.** We prove
Russo-Seymour-Welsh-type uniform bounds on crossing
probabilities for the FK Ising (FK percolation with
cluster weight $q = 2$) model at criticality, independent
of the boundary conditions. Our proof relies mainly on
Smirnov’s fermionic observable for the FK Ising model,
which allows us to get precise estimates on boundary
connection probabilities. We stay in a discrete setting,
in particular we do not make use of any continuum limit,
and our result can be used to derive directly several
noteworthy properties – including some new ones – among
which the fact that there is no inﬁnite cluster at
criticality, tightness properties for the interfaces, and
the existence of several critical exponents, in particular
the half-plane one-arm exponent. Such crossing bounds are
also instrumental for important applications such as
constructing the scaling limit of the Ising spin ﬁeld, and
deriving polynomial bounds for the mixing time of the
Glauber dynamics at criticality.

**Comment.** This version differs slightly from the
published version. A mistake is fixed in the proof of
Proposition 5.11. We thank C. Garban for bringing the
mistake to our attention and for useful discussions.

2010

**Bridge decomposition of Restriction Measures**

with T. Alberts, Journal of Statistical Physics, 140, 467-493, 2010.

**Abstract.** Motivated by
Kesten’s bridge decomposition for two-dimensional
self-avoiding walks in the upper half plane, we show
that the conjectured scaling limit of the half-plane
SAW, the SLE(8/3) process, also has an appropriately
deﬁned bridge decomposition. This continuum
decomposition turns out to entirely be a consequence
of the restriction property of SLE(8/3), and as a
result can be generalized to the wider class of
restriction measures. Speciﬁcally we show that the
restriction hulls with index less than one can be
decomposed into a Poisson Point Process of irreducible
bridges in a way that is similar to Ito’s excursion
decomposition of a Brownian motion according to its
excursions.

© Tom Alberts

**Parafermionic observables and their applications to planar statistical physics models.**

Ensaios Matematicos, Brazilian Mathematical Society, vol 25, 2013.

**A precision:** In the definition
p330 of $\mathcal E_\Omega(c,d)$, the interface from
$c$ to $d$ is required to be edge-avoiding with other
loops, and the number of loops includes the interface.

**Erratum:** Mistake in the formula of $\delta_x$
in (6.4) coming from a mistake in the second displayed
equation in p160. See the original article for the
relevant modification.

**Order/disorder phase transitions: the example of the Potts model.**

Lectures notes of the conference Current Developments in mathematics, 2015.

**Abstract.** Critical phenomena
at an order/disorder phase transition has been a
central object of interest in statistical physics. In
the past century, techniques borrowed from many
different fields of mathematics (Algebra,
Combinatorics, Probability, Complex Analysis, Spectral
Theory, etc) have contributed to a more and more
elaborated description of the possible critical
behaviors for a large variety of models (interacting
particle systems, lattice spin models, spin glasses,
percolation models). Through the classical examples of
the Ising and Potts models, we survey a few recent
advances regarding the rigorous understanding of such
phase transitions for the specific case of lattice
spin models. This review was written at the occasion
of the Harvard/MIT conference Current Developments in
Mathematics 2015.

© Vincent Beffara

**Graphical representations of lattice spin models.**

Lecture notes of Cours Peccot du Collège de France, Éditions Spartacus, 2015.

**Comment.** This book presents the
content of the Cours Peccot du Collège de France 2015 by
the author.

**Lectures on Planar percolation with a glimpse of Schramm Loewner Evolution.**

with V. Beffara, Probability Surveys, 10, 1-50, 2013.

**Abstract.** In recent years,
important progress has been made in the field of
two-dimensional statistical physics. One of the most
striking achievements is the proof of the
Cardy–Smirnov formula. This theorem, together with the
introduction of Schramm–Loewner Evolution and
techniques developed over the years in percolation,
allow precise descriptions of the critical and
near-critical regimes of the model. This survey aims
to describe the different steps leading to the proof
that the infinite-cluster density $\theta(p)$ for site
percolation on the triangular lattice behaves like
$(p−p_c)^{5/36+o(1)}$ as $p$ decreases to $1/2$.

** Comment.** Course given jointly with V. Beffara
at the La Pietra summer school 2011.

**Critical point and duality in planar lattice models.**

with V. Beffara, Probability and Statistical Physics in St. Petersburg, Editors Vladas Sidoravicius and Stanislav Smirnov, Proceedings of Symposia in Pure Mathematics, Amer. Math. Soc., 91, 2016.

**Abstract.** These lecture
notes describe the content of a six-hours course given
by the two authors at the 2012 probability summer
school in Saint-Petersburg. The goal is to provide a
derivation of several critical parameters of classical
planar models such as Bernoulli and Fortuin-Kasteleyn
percolation as well as the Ising and Potts models.

**Comment.** Course given jointly with V. Beffara
at the St Petersburg summer School 2012.

**Conformal invariance of lattice models.**

with S. Smirnov, Probability and Statistical Physics in Two and More Dimensions, Editors David Ellwood, Charles Newman, Vladas Sidoravicius, Wendelin Werner, Clay Mathematics Proceedings, vol. 15, Amer. Math. Soc., Providence, RI, 2012.

**Abstract.** These lecture notes
provide a (almost) self-contained account on conformal
invariance of the planar critical Ising and FK-Ising
models. They present the theory of discrete holomorphic
functions and its applications to planar statistical
physics (more precisely to the convergence of fermionic
observables). Convergence to SLE is discussed briefly.
Many open questions are included.

**Lectures on self-avoiding-walks.**

with R. Bauerschmidt, J. Goodman, and G. Slade, Probability and Statistical Physics in Two and More Dimensions, Editors David Ellwood, Charles Newman, Vladas Sidoravicius, Wendelin Werner, Clay Mathematics Proceedings, vol. 15, Amer. Math. Soc., Providence, RI, 2012.

**Abstract.**

These lecture notes provide a rapid
introduction to a number of rigorous results on
self-avoiding walks, with emphasis on the critical
behaviour. Following an introductory overview of the
central problems, an account is given of the
Hammersley--Welsh bound on the number of self-avoiding
walks and its consequences for the growth rates of
bridges and self-avoiding polygons. A detailed proof
that the connective constant on the hexagonal lattice
equals $\sqrt{2+\sqrt 2}$ is then provided. The lace
expansion for self-avoiding walks is described, and
its use in understanding the critical behaviour in
dimensions $d>4$ is discussed. Functional integral
representations of the self-avoiding walk model are
discussed and developed, and their use in a
renormalisation group analysis in dimension 4 is
sketched. Problems and solutions from tutorials are
included.

Vincent Beffara

**Introduction to percolation theory.**

**Abstract.** These lecture notes
present the content of a 10 hours class given for the
Master 2 of Paris-Saclay.

**Random currents expansion of the Ising model**

Proceedings of the 7th European Congress of Mathematicians in Berlin , 2016.

**Abstract.** Critical behavior at
an order/disorder phase transition has been a central
object of interest in statistical physics. In the past
century, techniques borrowed from many different fields of
mathematics (Algebra, Combinatorics, Probability, Complex
Analysis, Spectral Theory, etc) have contributed to a more
and more elaborate description of the possible critical
behaviors for a large variety of models. The Ising model
is maybe one of the most striking success of this
cross-fertilization, for this model of ferromagnetism is
now very well understood both physically and
mathematically. In this article, we review an approach,
initiated in an article of Griffiths, Hurst and Sherman
and based on the notion of random currents, enabling a
deep study of the model.

**RSW and Box-Crossing Property for Planar Percolation.**

with V. Tassion, IAMP proceedings , 2015.

**Abstract.** This article provides
a brief summary on recent advances on the so-called
Russo-Seymour-Welsh (RSW) Theory and its applications to
the study of planar percolation models. In particular, we
introduce a few properties of percolation models and
discuss their connections.

**A proof of first order phase transition for the planar random-cluster and Potts models with $q\gg1$.**

Proceedings of Stochastic Analysis on Large Scale Interacting Systems in RIMS kokyuroku Besssatu , 2016.

**Abstract.**We provide a proof
that the random-cluster model on the square lattice
undergoes a discontinuous phase transition for large
values of the cluster-weight $q$. This implies
discontinuity of the phase transition for Potts model
on the square lattice provided that the number of
colors $q$ is large enough. Let us remind the reader
that this result is classical and that we simply
provide an alternative approach based on the loop
representation.

**Parafermionic observables and their applications.**

IAMP bulletin , January 2016.

**Comment.** Short survey article
for the bulletin of the International Association of
Mathematical Physics.

**Phase transition in random-cluster and O(n)-models.**

PhD Thesis, 2011.

**Abstract.** This manuscript
gathers part of the work done during my phD. Several
chapters have been rewritten in order to use uniform
notation. An exposition of basic properties of the
random-cluster and $O(n)$ models has also been included to
make the manuscript as self-contained as possible.

**Law of the Iterated Logarithm for the random walk on the infinite percolation cluster.**

Master Thesis, 2008.

**Comment.** This is a research
project done when I was in UBC. The random walk on the
infinite supercritical percolation clusters in
$\mathbb{Z}^d$ is shown to satisfy the usual Law of the
Iterated Logarithm. The proof combines Barlow's Gaussian
heat kernel estimates and the ergodicity of the random
walk on the environment viewed from the random walker as
derived by Berger and Biskup.

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