arXiv subjects

Trishen S. Gunaratnam

Publications and source records attributed to Trishen S. Gunaratnam.

11 recordsLinked to original sources

arXiv2026

Supercritical sharpness for Ising double random currents

We show that the percolation model induced by the double random current representation of the Ising model on $\mathbb Z^d$, with $d\geq 3$, satisfies local uniqueness of the macroscopic cluster with high probability for every $β>β_c$, uniformly in boundary conditions (i.e. sources). Such a result is sometimes called supercritical sharpness in percolation theory and yields a detailed description of the model via standard renormalisation techniques. As applications, we show the exponential decay of truncated correlations for the Ising model and exponential mixing for the FK-Ising measure, results previously obtained by Duminil-Copin, Goswami and Raoufi. However, our proof is considerably simpler, as it replaces the use of a highly technical multi-valued map principle with a simple sprinkling argument.

math.PR↗

arXiv2026

Supercritical sharpness for the random cluster representation of real-valued spin models

We study the supercritical regime $β>β_c$ of a large family of real-valued spin models on $\mathbb Z^d$, including the Blume-Capel and $P(φ)$ 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 $φ^4$ models.

math.PR↗

arXiv2026

2d Sinh-Gordon model on the infinite cylinder

For $R>0$, we give a rigorous probabilistic construction on the cylinder $\mathbb{R} \times (\mathbb{R}/(2πR\mathbb{Z}))$ of the (massless) Sinh-Gordon model. In particular we define the $n$-point correlation functions of the model and show that these exhibit a scaling relation with respect to $R$. The construction, which relies on the massless Gaussian Free Field, is based on the spectral analysis of a quantum operator associated to the model. Using the theory of Gaussian multiplicative chaos, we prove that this operator has discrete spectrum and a strictly positive ground state.

math.PR↗

arXiv2025

The Markov property for $φ^4_3$ on the cylinder

We prove that the $φ^4_3$ model satisfies a version of Segal's axioms in the special case of three-dimensional tori and cylinders. As a consequence, we give the first proof that this model satisfies a Markov property and we characterize its boundary law up to absolutely continuous perturbations. In addition, we use Segal's axioms to give an alternative construction of the $φ^4_3$ Hamiltonian on two-dimensional tori as compared with Glimm (Comm. Math. Phys., 1968). We exploit this probabilistic approach to prove novel fundamental spectral properties of the Hamiltonian, such as discrete spectrum and a Perron-Froebenius type result on its ground state. The key technical contributions of this article are the development of tools to analyze $φ^4_3$ models with rough boundary conditions. We heavily use the variational approach to $φ^4_3$ models introduced in Barashkov and Gubinelli (Duke, 2020) that is based on the Boué-Dupuis formula and dual to Polchinski's continuous renormalization group.

math.PR↗

arXiv2025

The supercritical phase of the $φ^4$ model is well behaved

In this article, we analyse the $φ^4$ model on $\mathbb Z^d$ in the supercritical regime $β> β_c$. We consider a random cluster representation of the $φ^4$ 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 $β>β_c$, local uniqueness of macroscopic clusters occurs with high probability, uniformly in the 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 $φ^4$ models in the entire supercritical regime.

math.PR↗

arXiv2025

Random tangled currents for $φ^4$: translation invariant Gibbs measures and continuity of the phase transition

We prove that the set of automorphism invariant Gibbs measures for the $φ^4$ model on graphs of polynomial growth has at most two extremal measures at all values of $β$. 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 $φ^4$ 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 $φ^4$ model called the random tangled current representation.

math.PR↗

arXiv2024

Existence of a tricritical point for the Blume-Capel model on $\mathbb{Z}^d$

We prove the existence of a tricritical point for the Blume-Capel model on $\mathbb{Z}^d$ for every $d\geq 2$. The proof in $d\geq 3$ relies on a novel combinatorial mapping to an Ising model on a larger graph, the techniques of Aizenman, Duminil-Copin, and Sidoravicious (Comm. Math. Phys, 2015), and the celebrated infrared bound. In $d=2$, the proof relies on a quantitative analysis of crossing probabilities of the dilute random cluster representation of the Blume-Capel. In particular, we develop a quadrichotomy result in the spirit of Duminil-Copin and Tassion (Moscow Math. J., 2020), which allows us to obtain a fine picture of the phase diagram in $d=2$, including asymptotic behaviour of correlations in all regions. Finally, we show that the techniques used to establish subcritical sharpness for the dilute random cluster model extend to any $d\geq 2$.

math.PR↗

arXiv2024

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 $J_{x,y}=|x-y|^{-d-\boldsymbolα}$, where $\boldsymbolα>0$ 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.

math.PR↗

arXiv2023

Multiscale coupling and the maximum of $\mathcal{P}(ϕ)_2$ models on the torus

We establish a coupling between the $\mathcal{P}(ϕ)_2$ measure and the Gaussian free field on the two-dimensional unit torus at all spatial scales, quantified by probabilistic regularity estimates on the difference field. Our result includes the well-studied $ϕ^4_2$ measure. The proof uses an exact correspondence between the Polchinski renormalisation group approach, which is used to define the coupling, and the Boué-Dupuis stochastic control representation for $\mathcal{P}(ϕ)_2$. More precisely, we show that the difference field is obtained from a specific minimiser of the variational problem. This allows to transfer regularity estimates for the small-scales of minimisers, obtained using discrete harmonic analysis tools, to the difference field. As an application of the coupling, we prove that the maximum of the $\mathcal{P}(ϕ)_2$ field on the discretised torus with mesh-size $ε> 0$ converges in distribution to a randomly shifted Gumbel distribution as $ε\rightarrow 0$.

math.PR↗

arXiv2021

Quasi-invariant Gaussian measures for the nonlinear wave equation in three dimensions

We prove quasi-invariance of Gaussian measures supported on Sobolev spaces under the dynamics of the three-dimensional defocusing cubic nonlinear wave equation. As in the previous work on the two-dimensional case, we employ a simultaneous renormalization on the energy functional and its time derivative. Two new ingredients in the three-dimensional case are (i) the construction of the weighted Gaussian measures, based on a variational formula for the partition function inspired by Barashkov and Gubinelli (2018), and (ii) an improved argument in controlling the growth of the truncated weighted Gaussian measures, where we combine a deterministic growth bound of solutions with stochastic estimates on random distributions.

math.PR↗

arXiv2020

Phase transitions for $ϕ^4_3$

We establish a surface order large deviation estimate for the magnetisation of low temperature $ϕ^4_3$. As a byproduct, we obtain a decay of spectral gap for its Glauber dynamics given by the $ϕ^4_3$ singular stochastic PDE. Our main technical contributions are contour bounds for $ϕ^4_3$, which extends 2D results by Glimm, Jaffe, and Spencer (1975). We adapt an argument by Bodineau, Velenik, and Ioffe (2000) to use these contour bounds to study phase segregation. The main challenge to obtain the contour bounds is to handle the ultraviolet divergences of $ϕ^4_3$ whilst preserving the structure of the low temperature potential. To do this, we build on the variational approach to ultraviolet stability for $ϕ^4_3$ developed recently by Barashkov and Gubinelli (2019).

math.PR↗