arXiv subjects

A. Mazel

Publications and source records attributed to A. Mazel.

14 recordsLinked to original sources

arXiv2026

A Reverse Hard-Core Model in $\mathbb{A}_2$: an Application of the Pirogov-Sinai Theory

In this paper we use the Pirogov--Sinai theory to analyze a class of particle models of Statistical Mechanics on the unit triangular lattice $\mathbb{A}_2$. The models are specified by two parameters: the particle activity $u\in(0,1)$ and a positive L\"oschian number $t$ interpreted as a maximal squared clearing radius. Admissible configurations are those in which every empty lattice site has an occupied site within squared distance at most $t$, thereby forbidding empty disks of radius exceeding or equal $t$. The Hamiltonian favors configurations with as few occupied sites as possible, while the admissibility condition enforces a positive density of particles. We prove that the periodic ground states of the model, i.e., the configurations achieving an optimal balance of two tendencies, are the ones whose occupied sites form triangular sublattices of explicitly determined squared side-length $d_*(t)$. Furthermore, for sufficiently small values of the activity parameter $u$, some (but not necessarily all) periodic ground states generate extreme Gibbs (DLR) measures. We describe the resulting phase structure and characterize the pure phases associated with stable ground states.

math-ph↗

arXiv2026

A Nearest-Neighbor Hard-Core Model on a Penrose Graph

We prove that the maximal graph-density of an independent set in a Penrose P3 tiling considered as a planar non-directed graph is equal to $(57 - 25 \sqrt{5})/2 \approx 0.54915$ despite the fact that the graph is bipartite. Accordingly, the extreme Gibbs measure of the nearest-neighbor hard core particle model on this graph is unique for sufficiently large values of the particle activity. This invalidates a natural expectation to observe the coexistence of even and odd phases.

math-ph↗

arXiv2024

The Pirogov-Sinai Theory for Infinite Interactions

The purpose of this note is to consider a number of straightforward generalizations of the Pirogov-Sinai theory which can be covered by minor additions to the canonical texts. These generalizations are well-known among the adepts of the Pirogov-Sinai theory but are lacking formal references.

math-ph↗

arXiv2023

The hard-core model on $\mathbb{Z}^3$ and Kepler's conjecture

We study the hard-core model of statistical mechanics on a unit cubic lattice $\mathbb{Z}^3$, which is intrinsically related to the sphere-packing problem for spheres with centers in $\mathbb{Z}^3$. The model is defined by the sphere diameter $D>0$ which is interpreted as a Euclidean exclusion distance between point particles located at spheres centers. The second parameter of the underlying model is the particle fugacity $u$. For $u>1$ the ground states of the model are given by the dense-packings of the spheres. The identification of such dense-packings is a considerable challenge, and we solve it for $D^2=2, 3, 4, 5, 6, 8, 9, 10, 11, 12$ as well as for $D^2=2\ell^2$, where $\ell\in\mathbb{N}$. For the former family of values of $D^2$ our proofs are self-contained. For $D^2=2\ell^2$ our results are based on the proof of Kepler's conjecture. Depending on the value of $D^2$, we encounter three physically distinct situations: (i) finitely many periodic ground states, (ii) countably many layered periodic ground states and (iii) countably many not necessarily layered periodic ground states. For the first two cases we use the Pirogov-Sinai theory and identify the corresponding periodic Gibbs distributions for $D^2=2,3,5,8,9,10,12$ and $D^2=2\ell^2$, $\ell\in\mathbb{N}$, in a high-density regime $u>u_*(D^2)$, where the system is ordered and tends to fluctuate around some ground states. In particular, for $D^2=5$ only a finite number out of countably many layered periodic ground states generate pure phases.

math-ph↗

arXiv2022

Minimal Area of a Voronoi Cell in a Packing of Unit Circles

We present a new self-contained proof of the well-known fact that the minimal area of a Voronoi cell in a unit circle packing is equal to $2\sqrt{3}$, and the minimum is achieved only on a perfect hexagon. The proof is short and, in our opinion, instructive.

math.MG↗

arXiv2021

Kepler's conjecture and phase transitions in the high-density hard-core model on $\mathbb{Z}^3$

We perform a rigorous study of the identical sphere packing problem in $\mathbb{Z}^3$ and of phase transitions in the corresponding hard-core model. The sphere diameter $D>0$ and the fugacity $u\gg 1$ are the varying parameters of the model. We solve the sphere packing problem for values $D^2= 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 2\ell^2$, $\ell\in\mathbb{N}$. For values $D^2=2, 3, 5, 8, 9, 10, 12, 2\ell^2$, $\ell\in\mathbb{N}$ and $u>u^0(D)$ we establish the diagram of periodic pure phases, completely or partially. For the case $D^2=2\ell^2$, $\ell\in\mathbb{N}$ we use results from Hales' proof of Kepler's conjecture.

math-ph↗

arXiv2020

The hard-core model on planar lattices: the disk-packing problem and high-density phases

We study dense packings of disks and related Gibbs distributions representing high-density phases in the hard-core model on unit triangular, honeycomb and square lattices. The model is characterized by a Euclidean exclusion distance $D>0$ and a value of fugacity $u>0$. We use the Pirogov-Sinai theory to study the Gibbs distributions for a general $D$ when $u$ is large: $u>u_0(D)$. For infinite sequences of values $D$ we describe a complete high-density phase diagram: it exhibits a multitude of co-existing pure phases, and their number grows as $O(D^2)$. For the remaining values of $D$, except for those with sliding, the number of co-existing pure phases is still of the form $E(D)\geq O(D^2)$; however, the exact identification of the pure phases requires an additional analysis. Such an analysis is performed for a number of typical examples, which involves computer-assisted proofs. Consequently, for all values $D>0$ where sliding does not occur, we establish the existence of a phase transition. The crucial steps in the study are (i) the identification of periodic ground states and (ii) the verification of the Peierls bound. This is done by using connections with algebraic number theory. In particular, a complete list of so-called sliding values of $D$ has been specified. As a by-product, we solve the disk-packing problem on the lattices under consideration. The number and structure of maximally-dense packings depend on the disk-diameter $D$, unlike the case of $\mathbb{R}^2$.

math-ph↗

arXiv2019

High-density hard-core model on $\mathbb{Z}^2$ and norm equations in ring $\mathbb{Z} [{\sqrt[6]{-1}}]$

We study the Gibbs statistics of high-density hard-core configurations on a unit square lattice $\mathbb{Z}^2$, for a general Euclidean exclusion distance $D$. As a by-product, we solve the disk-packing problem on $\mathbb{Z}^2$ for disks of diameter $D$. The key point is an analysis of solutions to norm equations in $\mathbb{Z}[{\sqrt[6]{-1}}]$. We describe the ground states in terms of M-triangles, i.e., non-obtuse $\mathbb{Z}^2$-triangles of a minimal area with the side-lengths $\geq D$. There is a finite class (Class S) formed by values $D^2$ generating sliding, a phenomenon leading to countable families of periodic ground states. We identify all $D^2$ with sliding. Each of the remaining classes is proven to be infinite; they are characterized by uniqueness or non-uniqueness of a minimal triangle for a given $D^2$, up to $\mathbb{Z}^2$-congruencies. For values of $D^2$ with uniqueness (Class A) we describe the periodic ground states as admissible sub-lattices in $\mathbb{Z}^2$ of maximum density. By using the Pirogov-Sinai theory, it allows us to identify the extreme Gibbs measures (pure phases) for large values of fugacity and describe symmetries between them. Next, we analyze the values $D^2$ with non-uniqueness. For some $D^2$ all M-triangles are ${\mathbb{R}}^2$-congruent but not $\mathbb{Z}^2$-congruent (Class B0). For other values of $D^2$ there exist non-${\mathbb{R}}^2$-congruent M-triangles, with different collections of side-lengths (Class B1). Moreover, there are values $D^2$ for which both cases occur (Class B2). The large-fugacity phase diagram for Classes B0, B1, B2 is determined by dominant ground states. Classes A, B0-B2 are described in terms of cosets in $\mathbb{Z}[{\sqrt[6]{-1}}]$ by the group of units.

math.PR↗

arXiv2018

High-density hard-core model on triangular and hexagonal lattices

We perform a rigorous study of the Gibbs statistics of high-density hard-core random configurations on a unit triangular lattice $\mathbb{A}_2$ and a unit honeycomb graph $\mathbb{H}_2$, for any value of the (Euclidean) repulsion diameter $D>0$. Only attainable values of $D$ are relevant, for which $D^2=a^2+b^2+ab$, $a, b \in\mathbb{Z}$ (L\"oschian numbers). Depending on arithmetic properties of $D^2$, we identify, for large fugacities, the pure phases (extreme Gibbs measures) and specify their symmetries. The answers depend on the way(s) an equilateral triangle of side-length $D$ can be inscribed in $\mathbb{A}_2$ or $\mathbb{H}_2$. On $\mathbb{A}_2$, our approach works for all attainable $D^2$; on $\mathbb{H}_2$ we have to exclude $D^2 = 4, 7, 31, 133$, where a sliding phenomenon occurs, similar to that on a unit square lattice $\mathbb{Z}^2$. For all values $D^2$ apart from the excluded ones we prove the existence of a first-order phase transition where the number of co-existing pure phases grows at least as $O(D^2)$. The proof is based on the Pirogov--Sinai theory which requires non-trivial verifications of key assumptions: finiteness of the set of periodic ground states and the Peierls bound. To establish the Peierls bound, we develop a general method based on the concept of a re-distributed area for Delaunay triangles. Some of the presented proofs are computer-assisted. As a by-product of the ground state identification, we solve the disk-packing problem on $\mathbb{A}_2$ and $\mathbb{H}_2$ for any value of the disk diameter $D$.

math.PR↗

arXiv2014

A classical WR model with $q$ particle types

A version of the Widom--Rowlinson model is considered, where particles of $q$ types coexist, with a given collection of hard-core exclusion diameters. For $q\leq 4$, in the case of large equal fugacities, we give a complete description of the pure phase picture, based on the theory of dominant ground states.

math-ph↗

arXiv2014

Dominance of most tolerant species in multi-type lattice Widom-Rowlinson models

We analyse equilibrium phases in a multi-type lattice Widom-Rowlinson model with (i) four particle types, (ii) varying exclusion diameters between different particle types and (iii) large values of fugacity. Contrary to an expectation, it is not the most "aggressive" species, with largest diameters, which dominates the equilibrium measure, but the "most tolerant" one, which has smallest exclusion diameters. Results of numerical simulations are presented, showing densities of species in equilibrium phases and confirming the theoretical picture.

cond-mat.stat-mech↗

arXiv1998

Improved Peierls Argument for High Dimensional Ising Models

We consider the low temperature expansion for the Ising model on $\Z^d$, $d \ge 2$, with ferromagnetic nearest neighbor interactions in terms of Peierls contours. We prove that the expansion converges for all temperatures smaller than $C d (\log d)^{-1}$, which is the correct order in $d$.

cond-mat↗

arXiv1998

Liquid-Vapor Phase Transitions for Systems with Finite Range Interactions

We consider particles in $\R^d, d \geq 2$ interacting via attractive pair and repulsive four-body potentials of the Kac type. Perturbing about mean field theory, valid when the interaction range becomes infinite, we prove rigorously the existence of a liquid-gas phase transition, when the interaction range is finite but long compared to the interparticle spacing for a range of temperature.

cond-mat↗

arXiv1997

Ordering and Demixing Transitions in Multicomponent Widom-Rowlinson Models

We use Monte Carlo techniques and analytical methods to study the phase diagram of multicomponent Widom-Rowlinson models on a square lattice: there are M species all with the same fugacity z and a nearest neighbor hard core exclusion between unlike particles. Simulations show that for M between two and six there is a direct transition from the gas phase at z < z_d (M) to a demixed phase consisting mostly of one species at z > z_d (M) while for M \geq 7 there is an intermediate ``crystal phase'' for z lying between z_c(M) and z_d(M). In this phase, which is driven by entropy, particles, independent of species, preferentially occupy one of the sublattices, i.e. spatial symmetry but not particle symmetry is broken. The transition at z_d(M) appears to be first order for M \geq 5 putting it in the Potts model universality class. For large M the transition between the crystalline and demixed phase at z_d(M) can be proven to be first order with z_d(M) \sim M-2 + 1/M + ..., while z_c(M) is argued to behave as μ_{cr}/M, with μ_{cr} the value of the fugacity at which the one component hard square lattice gas has a transition, and to be always of the Ising type. Explicit calculations for the Bethe lattice with the coordination number q=4 give results similar to those for the square lattice except that the transition at z_d(M) becomes first order at M>2. This happens for all q, consistent with the model being in the Potts universality class.

cond-mat.stat-mech↗