Publications and preprints
[13]
One-arm exponents of the high-dimensional Ising model
with D. van Engelenburg, C. Garban, and F. Severo,
Preprint arXiv 2510.23423 (55 pages)
We study the probability that the origin is connected to the boundary of the box of size \(n\) (the one-arm probability) in several percolation models related to the Ising model. We prove that different universality classes emerge at criticality:
\(\bullet\) For the FK-Ising measure in a box of size \(n\) with wired boundary conditions, we prove that this probability decays as \(1/n\) in dimensions \(d>4\), and as \(1/n^{1+o(1)}\) when \(d=4\).
\(\bullet\) For the infinite volume FK-Ising measure, we prove that this probability decays as \(1/n^{2}\) in dimensions \(d>6\), and as \(1/n^{2+o(1)}\) when \(d=6\).
\(\bullet\) For the sourceless double random current measure, we prove that this probability decays as \(1/n^{d-2}\) in dimensions \(d>4\).
Additionally, for the infinite volume FK-Ising measure, we show that the one-arm probability is \(1/n^{1+o(1)}\) in dimension \(d=4\), and at least \(1/n^{3/2}\) in dimension \(d=5\). This establishes that the FK-Ising model has upper-critical dimension equal to \(6\),
in contrast to the Ising model, where it is known to be less or equal to \(4\), thus solving a conjecture of Chayes, Coniglio, Machta, and Shtengel.
Joint work with Diederik van Engelenburg, Christophe Garban, and Franco Severo.
[12]
One-arm exponents of high-dimensional percolation revisited
with D. van Engelenburg, C. Garban, and F. Severo,
Preprint arXiv 2510.21595 (16 pages)
We consider sufficiently spread-out Bernoulli percolation in dimensions \(d>6\). We present a short and simple proof of the up-to-constants estimate for the one-arm probability in both the full-space and half-space settings.
These results were previously established by Kozma and Nachmias and by Chatterjee and Hanson, respectively. Our proof improves upon the entropic technique introduced by Dewan and Muirhead,
relying on a sharp estimate on a suitably chosen correlation length recently obtained by Duminil-Copin and Panis. This approach is inspired by our companion work,
where we compute the one-arm exponent for several percolation models related to the high-dimensional Ising model.
Joint work with Diederik van Engelenburg, Christophe Garban, and Franco Severo.
[11]
The supercritical phase of the \(\varphi^4\) model is well behaved
with T.S. Gunaratnam, C. Panagiotis, and F. Severo,
Preprint arXiv 2501.05353 (79 pages)
In this article, we analyse the \(\varphi^4\) model on \(\mathbb{Z}^d\) in the supercritical regime \(\beta>\beta_c\).
We consider a random cluster representation of the \(\varphi^4\) model, which corresponds to an Ising random cluster model on a random environment.
We prove that the supercritical phase of this percolation model on \(\mathbb{Z}^d\) (\(d\geq 2\)) is well behaved in the sense that, for every \(\beta>\beta_c\),
local uniqueness of macroscopic clusters happens with high probability, uniformly in boundary conditions. This result provides the basis for renormalisation techniques
used to study several fine properties of the supercritical phase. As applications, we prove surface order exponential bounds for the (lower) large deviations of
the empirical magnetisation as well as for the spectral gaps of dynamical \(\varphi^4\) models in the entire supercritical regime.
Joint work with Trishen S. Gunaratnam, Christoforos Panagiotis, and Franco Severo.
[10]
An alternative approach for the mean-field behaviour of spread-out Bernoulli percolation in dimensions \(d > 6\)
with H. Duminil-Copin, arXiv 2410.03647, in Probability Theory and Related Fields (Special volume in honour of G.Grimmett) (2025)
This article proposes a new way of deriving mean-field exponents for sufficiently spread-out Bernoulli percolation in dimensions \(d > 6\).
We obtain up-to-constant estimates for the full-space and half-space two-point functions in the critical and near-critical regimes.
In a companion paper, we apply a similar analysis to the study of the weakly self-avoiding walk model in dimensions \(d > 4\).
Joint work with Hugo Duminil-Copin.
Here is a talk that I gave at Percolation Today.
[9]
An alternative approach for the mean-field behaviour of weakly self-avoiding walks in dimensions \(d>4\)
with H. Duminil-Copin, arXiv 2410.03649, in Probability Theory and Related Fields (Special volume in honour of G.Grimmett) (2025)
This article proposes a new way of deriving mean-field exponents for the weakly self-avoiding walk model in dimensions \(d>4\).
Among other results, we obtain up-to-constant estimates for the full-space and half-space two-point functions in the critical and near-critical regimes.
A companion paper proposes a similar analysis for spread-out Bernoulli percolation in dimensions \(d>6\).
Joint work with Hugo Duminil-Copin.
[8]
The incipient infinite cluster of the FK-Ising model in dimensions \(d\ge 3\) and the susceptibility of the high-dimensional Ising model
arXiv 2406.15243, in Probability Theory and Related Fields (2025)
We consider the critical FK-Ising measure \(\phi_{\beta_c}\) on \(\mathbb Z^d\) with \(d\geq 3\). We construct the measure \(\phi^\infty:=\lim_{|x|\rightarrow \infty}\phi_{\beta_c}[\:\cdot\: |\: 0\leftrightarrow x]\)
and prove it satisfies \(\phi^\infty[0\leftrightarrow\infty]=1\). This corresponds to the natural candidate for the
incipient infinite cluster measure of the FK-Ising model.
Our proof uses a result of Lupu and Werner (Electron. Commun. Probab., 2016) that relates the FK-Ising model to the random current representation of the Ising model,
together with a
mixing property of random currents recently established by Aizenman and Duminil-Copin (Ann. Math., 2021).
We then study the susceptibility \(\chi(\beta)\) of the nearest-neighbour Ising model on \(\mathbb Z^d\). When \(d>4\),
we improve a previous result of Aizenman (Comm. Math. Phys., 1982) to obtain the existence of \(A>0\) such that, for \(\beta<\beta_c\),
\[
\chi(\beta)= \frac{A}{1-\beta/\beta_c}(1+o(1)),
\]
where \(o(1)\) tends to \(0\) as \(\beta\) tends to \(\beta_c\).
Additionally, we relate the constant \(A\) to the incipient infinite cluster of the double random current.
[7]
The torus plateau for the high-dimensional Ising model
with Y. Liu and G. Slade,
arXiv 2405.17353, in Communications in Mathematical Physics (2025)
We consider the Ising model on a \(d\)-dimensional discrete torus of volume \(r^{d}\),
in dimensions \(d>4\) and for large \(r\), in the vicinity of the infinite-volume critical
point \(\beta_c\).
We prove that for \(\beta=\beta_c- {\rm const}\, r^{-d/2}\) (with a suitable
constant) the susceptibility is bounded above and below by multiples of \(r^{d/2}\),
and that the two-point function has a "plateau" in the sense that it decays
like \(|x|^{-(d-2)}\) when \(|x|\) is small relative to the volume but
for larger \(|x|\) it levels off to
a constant value of order \(r^{-d/2}\). We also prove that
at \(\beta=\beta_c- {\rm const}\, r^{-d/2}\) the renormalised
coupling constant is nonzero, which implies a non-Gaussian limit for the average
spin. The proof relies on near-critical estimates for the
infinite-volume two-point function obtained recently by Duminil-Copin and Panis,
and builds upon a strategy proposed by Papathanakos.
The random current representation of
the Ising model plays a central role in our analysis.
Joint work with Yucheng Liu and Gordon Slade.
[6]
New lower bounds for the (near) critical Ising and \(\varphi^{4}\) models' two-point functions
with H. Duminil-Copin,
arXiv 2404.05700, in Communications in Mathematical Physics (2025)
We study the nearest-neighbour Ising and \(\varphi^{4}\) models on \(\mathbb{Z}^{d}\) with \(d\geq 3\) and obtain new lower bounds on their two-point functions at (and near) criticality.
Together with the classical infrared bound, these bounds turn into up-to constant estimates when \(d\geq 5\). When \(d=4\), we obtain an ''almost'' sharp lower bound corrected by a logarithmic factor.
As a consequence of these results, we show that \(\eta=0\) and \(\nu=1/2\) when \(d\geq 4\), where \(\eta\) is the critical exponent associated with the decay of the model's two-point function at criticality
and \(\nu\) is the critical exponent of the correlation length \(\xi(\beta)\). When \(d=3\), we improve previous results and obtain that \(\eta\leq 1/2\).
As a byproduct of our proofs, we also derive the blow-up at criticality of the so-called bubble diagram when \(d=3,4\).
Joint work with Hugo Duminil-Copin.
[5]
Exact cube-root fluctuations in an area-constrained random walk model
with L. D'Alimonte,
Preprint arXiv 2311.12780 (48 pages)
This article is devoted to the study of the behaviour of a (1+1)-dimensional model of random walk conditioned to enclose an area of order \(N^2\).
Such a conditioning enforces a globally concave trajectory. We study the local deviations of the walk from its convex hull. To this end,
we introduce two quantities, the mean facet length \(\mathsf{MeanFL}\) and the mean local roughness \(\mathsf{MeanLR}\), measuring the typical longitudinal and transversal
fluctuations around the boundary of the convex hull of the random walk. Our main result is that \(\mathsf{MeanFL}\) is of order \(N^{2/3}\) and \(\mathsf{MeanLR}\) is of order \(N^{1/3}\). Moreover, following the strategy of
Hammond (Ann. Prob., 2012), we identify the polylogarithmic corrections in the scaling of the
maximal facet length and of the
maximal local roughness,
showing that the former one scales as \(N^{2/3}(\log N)^{1/3}\), while the latter scales as \(N^{1/3}(\log N)^{2/3}\). The object of study is intended to be a toy model for the
interface of a two-dimensional statistical mechanics model (such as the Ising model) in the phase separation regime, we discuss this issue at the end of this work.
Joint work with Lucas D'Alimonte.
[4]
Scaling limit of the cluster size distribution for the random current measure on the complete graph
with D. Krachun and C. Panagiotis,
arXiv 2310.02087, in Electronic Journal of Probability (2024)
We study the percolation configuration arising from the random current representation of the near-critical Ising model on the complete graph.
We compute the scaling limit of the cluster size distribution for an arbitrary set of sources in the single and
the double current measures. As a byproduct, we compute the tangling probabilities recently introduced by Gunaratnam, Panagiotis, Panis,
and Severo in [GPPS]. This provides a new perspective on the switching lemma for the \(\varphi^4\) model
introduced in the same paper: in the Gaussian limit we recover Wick's law, while in the Ising limit we recover
the corresponding tool for the Ising model.
Joint work with Dmitrii Krachun and Christoforos Panagiotis.
[3]
Triviality of the scaling limits of critical Ising and \(\varphi^4\) models with effective dimension at least four
Preprint arXiv 2309.05797 (86 pages), To appear in The Annals of Probability
We prove that any scaling limit of a critical reflection positive Ising or \(\varphi^4\) model of effective dimension \(d_{\text{eff}}\)
at least four is Gaussian. This extends the recent breakthrough work of Aizenman and Duminil-Copin, which
demonstrates the corresponding result in the setup of nearest-neighbour interactions in dimension four, to the case of long-range reflection
positive interactions satisfying \(d_{\text{eff}}=4\). The proof relies on the random current representation which provides a geometric interpretation of the deviation of the
models' correlation functions from Wick's law. When \(d=4\), long-range interactions are handled with the derivation of a criterion that relates the speed of decay of the interaction to
two different mechanisms that entail Gaussianity: interactions with a sufficiently slow decay induce a faster decay at the level of the model's two-point function, while sufficiently fast decaying interactions
force a simpler geometry on the currents which allows to extend nearest-neighbour arguments. When \(1\leq d\leq 3\) and \(d_{\text{eff}}=4\), the phenomenology is different as
long-range effects play a prominent role.
[2]
Emergence of fractional Gaussian free field correlations in subcritical long-range Ising models
with T.S. Gunaratnam,
arXiv 2306.11887, in Annales de l'Institut Henri Poincaré (B) (2025)
We study corrections to the scaling limit of subcritical long-range Ising models with (super)-summable interactions on \(\mathbb Z^d\).
For a wide class of models, the scaling limit is known to be white noise, as shown by Newman (1980).
In the specific case of couplings \(J_{x,y}=|x-y|^{-d-\boldsymbol{\alpha}}\), where \(\boldsymbol{\alpha}>0\) and \(|\cdot|\) is the Euclidean norm,
we find an emergence of fractional Gaussian free field correlations in appropriately renormalised and rescaled observables.
The proof exploits the exact asymptotics of the two-point function, first established by Newman and Spohn (1998), together with the rotational symmetry of the interaction.
Joint work with Trishen S. Gunaratnam.
[1]
Random tangled currents for \(\varphi^4\): translation invariant Gibbs measures and continuity of the phase transition
with T.S. Gunaratnam, C. Panagiotis, and F. Severo,
arXiv 2211.00319, in Journal of the European Mathematical Society (2025)
We prove that the set of automorphism invariant Gibbs measures for the \(\varphi^4\) model on graphs of polynomial
growth has at most two extremal measures at all values of \(\beta\). We also give a sufficient condition to ensure that
the set of all Gibbs measures is a singleton. As an application, we show that the spontaneous magnetisation of the
nearest-neighbour \(\varphi^4\) model on \(\mathbb{Z}^d\) vanishes at criticality for \(d\geq 3\).
The analogous results were established for the Ising model in the seminal works of Aizenman, Duminil-Copin, and Sidoravicius (Comm. Math. Phys., 2015),
and Raoufi (Ann. Prob., 2020) using the so-called random current representation introduced by Aizenman (Comm. Math. Phys., 1982). One of the main contributions of this
paper is the development of a corresponding geometric representation for the \(\varphi^4\) model
called the
random tangled current representation.
Joint work with Trishen S. Gunaratnam, Christoforos Panagiotis, and Franco Severo.
Here is a talk that Trishen and I gave at Percolation Today.