Mass scaling of the near-critical Ising model in dimensions \(d\geq 4\)
We study the Ising model on \(\mathbb{Z}^d\) with \(d\geq 4\) and derive near-critical bounds on the truncated two-point function \(\langle \sigma_0;\sigma_x\rangle_{\beta,h}-\langle\sigma_0\rangle_{\beta,h}\langle\sigma_x\rangle_{\beta,h}\) at
parameters \(\beta\leq \beta_c\) and \(h\geq 0\). As a corollary, we obtain that the associated mass (or exponential decay rate) is equal to \( \max((\beta_c-\beta)^{1/2},h^{1/3})^{1+o(1)}\), where \(o(1)\) tends to \(0\) as \((\beta,h)\) tends to \((\beta_c,0)\). The proof combines
the corresponding result at \(h=0\), recently established by Duminil-Copin and Panis, with an interpolation argument inspired by Aizenman and Fernandez and carried out via the random current representation of the model.
Supercritical sharpness for the random cluster representation of real-valued spin models
We study the supercritical regime \(\beta>\beta_c\) of a large family of real-valued spin models on \(\mathbb{Z}^d\), including in particular the Blume-Capel and \(P(\varphi)\) models.
We consider their random cluster representations and prove that they are well-behaved, in the sense that local uniqueness of macroscopic clusters occurs with high probability,
uniformly in the boundary conditions. This implies, among other things, a surface-order exponential bound for the (lower) large deviations of the empirical magnetisation.
These results were previously known only in the cases of the Ising and \(\varphi^4\) models.
Joint work with Trishen S. Gunaratnam, Dmitrii Krachnu, Christoforos Panagiotis, and Franco Severo.
On reversing the Simon-Lieb inequality in high-dimensional percolation
We study Bernoulli percolation on \(\mathbb Z^d\) in dimensions \(d>6\).
We prove that a classical consequence of the van den Berg-Kesten inequality, often referred to as the Simon-Lieb inequality
in the context of the Ising model, admits a partial reversal. As a main application, we show that the quantity \(\varphi_{p_c}(S)\),
introduced by Duminil-Copin and Tassion (Comm. Math. Phys., 2016), is uniformly bounded over all \(S\subset \mathbb Z^d\).
This partial reversal further yields a short and self-contained route to several key results, including near-critical estimates on the two-point function
and sharp bounds on the critical one-arm probability.
A random walk approach to high-dimensional critical phenomena
We present a "black box" proof of mean-field near-critical behaviour for
a family of functions on \(\mathbb Z^d\) (with \(d>2\)) satisfying a short list of assumptions.
The functions represent two-point functions of a lattice statistical mechanical
model in the subcritical or critical regimes, and are proved to have
decay of the form \(|x|^{-d+2+\varepsilon}\exp[-c|x|/\xi]\), for any \(\varepsilon>0\).
The black box applies to
several models for which commonplace methods can be used to verify the
assumptions.
Applications include models of self-avoiding walk, percolation, spins (Ising, XY, \(|\varphi|^4\)), and lattice trees, all above their upper critical dimensions.
The proof is based on random walk techniques, and
provides a new, unified, probabilistic,
and relatively simple proof of mean-field near-critical behaviour.
Joint work with Hugo Duminil-Copin, Aman Markar, and Gordon Slade.
Sharp bounds on the half-space two-point function for high-dimensional Bernoulli percolation
We consider Bernoulli percolation on \(\mathbb{Z}^d\) with \(d>6\). We prove an up-to-constant estimate
for the critical two-point function restricted to a half-space. This completes previous results
of Chatterjee and Hanson (Commun. Pure Appl. Math., 2021), and Chatterjee, Hanson, and
Sosoe (Commun. Math. Phys., 2022), and solves a question asked by Hutchcroft, Michta, and
Slade (Ann. Probab., 2023).
One-arm exponents of the high-dimensional Ising model
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.
One-arm exponents of high-dimensional percolation revisited
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.
Random tangled currents for \(\varphi^4\): translation invariant Gibbs measures and continuity of the phase transition
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.
An alternative approach for the mean-field behaviour of spread-out Bernoulli percolation in dimensions \(d > 6\)
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 \(d > 4\).
An alternative approach for the mean-field behaviour of weakly self-avoiding walks in dimensions \(d>4\)
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 \(d>6\).
The incipient infinite cluster of the FK-Ising model in dimensions \(d\ge 3\) and the susceptibility of the high-dimensional Ising model
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.
The torus plateau for the high-dimensional Ising model
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.
New lower bounds for the (near) critical Ising and \(\varphi^{4}\) models' two-point functions
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 \(d=3,4\).
Exact cube-root fluctuations in an area-constrained random walk model
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.
Scaling limit of the cluster size distribution for the random current measure on the complete graph
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.
Triviality of the scaling limits of critical Ising and \(\varphi^4\) models with effective dimension at least four
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.
Emergence of fractional Gaussian free field correlations in subcritical long-range Ising models
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.
Random tangled currents for \(\varphi^4\): translation invariant Gibbs measures and continuity of the phase transition
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.
