Search arXiv⌕ Search

arXiv subjects

Aldo Procacci

Publications and source records attributed to Aldo Procacci.

At least 19 recordsLinked to original sources
arXiv2026

The BEG model at the FAD triple point on the square lattice

In this note, we prove that the two-dimensional Blume-Emery-Griffiths model at the triple point Ferromagnetic-Antiquadrupolar-Disordered (FAD) has a unique Gibbs measure at any temperature, thereby establishing the absence of phase transitions. The FAD point lies at the intersection of lines separating three regions of the phase diagram, and it is a singular point where the model exhibits infinitely many ground states. Our proof is based on a random-cluster type representation with configuration-dependent cluster weights and a coupling with Bernoulli site percolation with parameter $1/2$.

math-ph↗
arXiv2026

On the zero-free region for the chromatic polynomial of claw-free graphs with and without induced square and induced diamond

Given a claw-free graph $G=(V,E)$ with maximum degree $\Delta$, we define the parameter $\kappa\in [0,1]$ as $\kappa={\max_{v\in V}|I_v|\over \lfloor\Delta^2/4\rfloor}$ where $I_v$ is the set of all independent pairs in the neighborhood of $v$. We refer to $\kappa$ as the pair independence ratio of $G$. We prove that for any claw-free graph $G$ with pair independence ratio at most $\kappa$ the zeros of its chromatic polynomial $P_G(q)$ lie inside the disk $D=\{q\in \mathbb{C}:~|q|< C_\kappa^0\Delta\}$, where $C_\kappa^0$ is an increasing function of $\kappa\in [0,1]$. If $G$ is also square-free and diamond free, the function $C_\kappa^0$ can be replaced by a sharper function $C_\kappa^1$. These bounds constitute an improvement upon results recently given by Bencs and Regts in ''Improved bounds on the zeros of the chromatic polynomial of graphs and claw-free graphs''.

math.CO↗
arXiv2025

On the independent set polynomial of graphs and claw-free graphs

We present two new contributions to the study of the independence polynomial $Z_G(z)$ of a finite simple graph $G = (V,E)$. First, we provide an improved lower bound for the zero-free region of $Z_G(z)$ for the important class of claw-free graphs. Our bound exceeds the classical Shearer radius and it is derived through a refined application of the Fern\'andez-Procacci criterion using properties of the local neighborhood structure in claw-free graphs. Second, we establish a novel combinatorial expression for $Z_G(z)$, inspired by the connection with the abstract polymer gas models in statistical mechanics, which offers a new structural interpretation of the polynomial and may be of independent interest. These results strengthen the connection between statistical physics, combinatorics, and graph theory, and suggest new approaches for analytic exploration.

math.CO↗
arXiv2024

On the zero-free region for the chromatic polynomial of graphs with maximum degree $\Delta$ and girth $g$

The purpose of the present paper is to provide, for all pairs of integers $(\Delta,g)$ with $\D\ge 3$ and $g\ge 3$, a positive number $C(\Delta, g)$ such that chromatic polynomial $P_G(q)$ of a graph $G$ with maximum degree $\Delta$ and finite girth $g$ is free of zero if $|q|\ge C(\Delta, g)$. Our bounds enlarge the zero-free region in the complex plane of $P_G(q)$ in comparison to previous bounds. In particular, for small values of $\D$ our estimates yield a sensible improvement on the bounds recently obtained by Jenssen, Patel and Regts in \cite{JPR}, while they coincide with those of \cite{JPR} when $\Delta\to \infty$.

math.CO↗
arXiv2024

A remark on the Whitney Broken Circuit Theorem

In the present note we show, via the connection between chromatic polynomial and Potts model, that the Whitney Broken circuit theorem is in fact a special case of a more general identity relating the chromatic polynomial of a graph G=(V,E) to sums over forests of G associated to some partition scheme in G.

math.CO↗
arXiv2024

Anisotropic Ising model in d+s dimensions

In this note, we consider the asymmetric nearest neighbor ferromagnetic Ising model on the $(d+s)$-dimensional unit cubic lattice $\Z^{d+s}$, at inverse temperature $\beta=1$ and with coupling constants $J_s>0$ and $J_d>0$ for edges of $\Z^s$ and $\Z^d$, respectively. We obtain a lower bound for the critical curve in the phase diagram of $(J_s,J_d)$. In particular, as $J_d$ approaches its critical value from below, our result is directly related to the so-called dimensional crossover phenomenon.

math-ph↗
arXiv2023

Cluster expansion methods in rigorous statistical mechanics

This draft is intended to be used as class notes for a grad course on rigorous statistical mechanics at math department of UFMG. It should be considered as a very prelimivary version and a work in progress. Several chapters lack references, exercises, and revision.

math-ph↗
arXiv2022

The Blume-Emery-Griffiths model at the FAD and AD interfaces

We analyse the Blume-Emery-Griffiths (BEG) model on the lattice $\Zd$ at the ferromagnetic-antiquadrupolar-disordered (FAD) and antiquadrupolar-disordered (AD) interfaces of parameters. In our analysis of the FAD interface we introduce a Gibbs sampler of the ground states at zero temperature, and we exploit it in two different ways: first, we perform via perfect sampling an empirical evaluation of the spontaneous magnetization at zero temperature, finding a non-zero value in $d=3$ and a vanishing value in $d=2$. Second, using a careful coupling with the Bernoulli site percolation model in $d=2$, we prove rigorously that imposing $+$ boundary conditions, the magnetization in the center of a square box tends to zero in the thermodynamical limit and the two-point correlations decay exponentially. Also, using again a coupling argument, we show that the infinite volume Gibbs measure of the zero-temperature BEG exists and it is unique. In our analysis of the AD interface we restrict ourselves to $d=2$ and, by comparing the BEG model with a Bernoulli site percolation in a matching graph of $\mathbb{Z}^2$, we get a condition for the vanishing of the infinite volume limit magnetization improving, for low temperatures, earlier results obtained via expansion techniques.

math-ph↗
arXiv2020

Classical particles in the continuum subjected to high density boundary conditions

We consider a continuous system of classical particles confined in a finite region $\Lambda$ of $\mathbb{R}^d$ interacting through a superstable and tempered pair potential in presence of non free boundary conditions. We prove that the thermodynamic limit of the pressure of the system at any fixed inverse temperature $\beta$ and any fixed fugacity $\lambda$ does not depend on boundary conditions produced by particles outside $\Lambda$ whose density may increase sub-linearly with the distance from the origin at a rate which depends on how fast the pair potential decays at large distances. In particular, if the pair potential $v(x-y)$ is of Lennard-Jones type, i.e. it decays as $C/\|x-y\|^{d+p}$ (with $p>0$) where $\|x-y\|$ is the Euclidean distance between $x$ and $y$, then the existence of the thermodynamic limit of the pressure is guaranteed in presence of boundary conditions generated by external particles which may be distributed with a density increasing with the distance $r$ from the origin as $\rho(1+ r^q)$, where $\rho$ is any positive constant (even arbitrarily larger than the density $\rho_0(\beta,\lambda)$ of the system evaluated with free boundary conditions) and $q\le {1\over 2}\min\{1, p\}$.

math-ph↗
arXiv2020

Virial series for a system of classical particles interacting through a pair potential with negative minimum

In this note we revisit the recent developments concerning rigorous results on the virial series of a continuous system of classical particles interacting via a stable and tempered pair potential and we provide new lower bounds for its convergence radius when the potential has a strictly positive stability constant. As an application we obtain a new estimate for the convergence radius of the virial series of the Lennard-Jones gas which improves sensibly previous estimates present in the literature.

math-ph↗
arXiv2019

A new bound on the acyclic edge chromatic index

In this note we obtain a new bound for the acyclic edge chromatic number $a'(G)$ of a graph $G$ with maximum degree $D$ proving that $a'(G)\leq 3.569(D-1)$. To get this result we revisit and slightly modify the method described in [Giotis, Kirousis, Psaromiligkos and Thilikos, Theoretical Computer Science, 66: 40-50, 2017].

math.CO↗
arXiv2017

A correction to a remark in a paper by Procacci and Yuhjtman: new lower bounds for the convergence radius of the virial series

In this note we deduce a new lower bound for the convergence radius of the Virial series of a continuous system of classical particles interacting via a stable and tempered pair potential using the estimates on the Mayer coefficients obtained in the recent paper by Procacci and Yuhjtman (Lett Math Phys 107:31-46, 2017). This corrects the wrongly optimistic lower bound for the same radius claimed (but not proved) in the above cited paper (in Remark 2 below Theorem 1). The lower bound for the convergence radius of the Virial series provided here represents a strong improvement on the classical estimate given by Lebowitz and Penrose in 1964.

math-ph↗
arXiv2017

Effects of boundary conditions on irreversible dynamics

We present a simple one-dimensional Ising-type spin system on which we define a completely asymmetric Markovian single spin-flip dynamics. We study the system at a very low, yet non-zero, temperature and we show that for empty boundary conditions the Gibbs measure is stationary for such dynamics, while introducing in a single site a $+$ condition the stationary measure changes drastically, with macroscopical effects. We achieve this result defining an absolutely convergent series expansion of the stationary measure around the zero temperature system. Interesting combinatorial identities are involved in the proofs.

math-ph↗