arXiv subjects

Nguyen Tong Xuan

Publications and source records attributed to Nguyen Tong Xuan.

4 recordsLinked to original sources

arXiv2026

New results on the domain of analyticity of the free energy for the Ising model

We investigate the analyticity of the free energy of the Ising model in the presence of a non-zero external magnetic field, at high temperature, and at low temperature. Using the Fernandez--Procacci convergence criterion for cluster expansions, together with generating-function techniques and graph-theoretical methods, we derive improved convergence conditions in all three regimes. In particular, the generating-function approach yields sharper estimates for polymers and contours in the strong-field and low-temperature regimes, while a new high-temperature expansion based on Veblen's theorem provides a substantially larger analyticity region than the classical results in the literature.

math-ph↗

arXiv2024

Local Central Limit Theorem for unbounded long-range potentials

We prove the equivalence between the integral central limit theorem and the local central limit theorem for two-body potentials with long-range interactions on the lattice $\mathbb{Z}^d$ for $d\ge 1$. The spin space can be an arbitrary, possibly unbounded subset of the real axis with a suitable a-priori measure. For general unbounded spins, our method works at high-enough temperature, but for bounded spins our results hold for every temperature. Our proof relies on the control of the integrated characteristic function, which is achieved by dividing the integration into three different regions, following a standard approach proposed forty years ago by Campanino, Del Grosso and Tirozzi. The bounds required in the different regions are obtained through cluster-expansion techniques. For bounded spins, the arbitrariness of the temperature is achieved through a decimation ("dilution") technique, also introduced in the later reference.

math-ph↗

arXiv2023

High-temperature cluster expansion for classical and quantum spin lattice systems with multi-body interactions

We develop a novel cluster expansion for finite-spin lattice systems subject to multi-body quantum -- and, in particular, classical -- interactions. Our approach is based on the use of ``decoupling parameters", advocated by Park [34], which relates partition functions with successive additional interaction terms. Our treatment, however, leads to an explicit expansion in a $β$-dependent effective fugacity that permits an explicit evaluation of free energy and correlation functions at small $β$. To determine its convergence region we adopt a relatively recent cluster summation scheme that replaces the traditional use of Kikwood-Salzburg-like integral equations by more precise sums in terms of particular tree-diagrams [2]. As an application we show that our lower bound of the radius of $β$-analyticity is larger than Park's for quantum systems two-body interactions.

math-ph↗

arXiv2019

Convergence of cluster and virial expansions for repulsive classical gases

We study the convergence of cluster and virial expansions for systems of particles subject to positive two-body interactions. Our results strengthen and generalize existing lower bounds on the radii of convergence and on the value of the pressure. Our treatment of the cluster coefficients is based on expressing the truncated weights in terms of trees and partition schemes, and generalize to soft repulsions previous approaches for models with hard exclusions. Our main theorem holds in a very general framework that does not require translation invariance and is applicable to models in general measure spaces. For the virial coefficients we resort to an approach due to Ramawadth and Tate that uses Lagrange inversion techniques only at the level of formal power series and leads to diagrammatic expressions in terms of trees, rather than the doubly connected diagrams traditionally used. We obtain a new criterion that strengthens, for repulsive interactions, the best criterion previously available (proposed by Groenveld and proven by Ramawadth and Tate). We illustrate our results with a few applications showing noticeable improvements in the lower bound of convergence radii.

math-ph↗