arXiv subjects

I. Stuhl

Publications and source records attributed to I. Stuhl.

17 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

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↗

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↗

arXiv2023

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↗

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↗

arXiv2022

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↗

arXiv2020

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 $D^2=a^2+b^2+ab$, $a, b \in\mathbb{Z}$ (Löschian 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↗

arXiv2017

Nilpotent Steiner loops of class 2

We describe Steiner loops of nilpotency class 2 and establish the classification of finite 3-generated nilpotent Steiner loops of nilpotency class 2.

math.GR↗

arXiv2016

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↗

arXiv2015

Weight functions and log-optimal investment portfolios

Following the paper by Algoet--Cover (1988), we analyse log-optimal portfolios where return evaluation includes `weights' of different outcomes. The results are twofold: (A) under certain conditions, logarithmic growth rate is a supermartingale, and (B) the optimal (martingale) investment strategy is a proportional betting; it does not depend on the form of the weight function, although the optimal rate does. The existence of an optimal investment strategy has been established earlier in a great generality by Kramkov--Schachermayer (2003) although our underlying assumptions are different.

math.PR↗

arXiv2015

Weighted Gaussian entropy and determinant inequalities

We produce a series of results extending information-theoretical inequalities (discussed by Dembo--Cover--Thomas in 1989-1991) to a weighted version of entropy. The resulting inequalities involve the Gaussian weighted entropy; they imply a number of new relations for determinants of positive-definite matrices.

cs.IT↗

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

Free Steiner triple systems and their automorphism groups

The paper is devoted to the study of free objects in the variety of Steiner loops and of the combinatorial structures behind them, focusing on their automorphism groups. We prove that all automorphisms are tame and the automorphism group is not finitely generated if the loop is more than $3$-generated. For the free Steiner loop with $3$ generators we describe the generator elements of the automorphism group and some relations between them.

math.GR↗

arXiv2013

FK-DLR properties of a quantum multi-type bose-gas with a repulsive interaction

The paper extends earlier results from \cite{SK}, \cite{SKS} about infinite-volume quantum bosonic states (FK-DLR states) to the case of multi-type particles with non-negative interactions. (An example is a quantum Widom--Rowlinson model.) Following the strategy from \cite{SK}, \cite{SKS}, we establish that, for the values of fugacity $z\in (0,1)$ and inverse temperature $β>0$, finite-volume Gibbs states form a compact family in the thermodynamic limit. Next, in dimension two we show that any limit-point state (an FK-DLR state in the terminology adopted in \cite{SK}, \cite{SKS}) is translation-invariant.

math-ph↗

arXiv2013

Shift-invariance for FK-DLR states of a 2D quantum bose-gas

This paper continues the work Y. Suhov, M. Kelbert. FK-DLR states of a quantum bose-gas, arXiv:1304.0782 [math-ph], and focuses on infinite-volume bosonic states for a quantum system (a quantum gas) in a plane. We work under similar assumptions upon the form of local Hamiltonians and the type of the (pair) interaction potential as in the reference above. The result of the paper is that any infinite-volume FK-DLR functional corresponding to the Hamiltonians is shift-invariant, regardless of whether this functional is unique or not.

math-ph↗