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H. J. Hilhorst

Publications and source records attributed to H. J. Hilhorst.

At least 19 recordsLinked to original sources
arXiv2024

Many-sided Poisson-Voronoi cells with only Gabriel neighbors

Let $p_n^G$ be the probability for a planar Poisson-Voronoi cell to be $n$-sided {\it and\,} have only Gabriel neighbors. Using an exact coordinate transformation followed by scaling arguments and a mean-field type calculation, we obtain the asymptotic expansion of $\log p_n^G$ in the limit of large $n$. We determine several statistical properties of a many-sided cell obeying this `Gabriel condition.' In particular, the cell perimeter, when parametrized as a function $τ(θ)$ of the polar angle $θ$, behaves as a Brownian bridge on the interval $0\leθ\le 2π$. We point out similarities and differences with related problems in random geometry.

cond-mat.stat-mech↗
arXiv2023

New duality relation for the Discrete Gaussian SOS model on a torus

We construct a new duality for two-dimensional Discrete Gaussian models. It is based on a known one-dimensional duality and on a mapping, implied by the Chinese remainder theorem, between the sites of an $N\times M$ torus and those of a ring of $NM$ sites. The duality holds for an arbitrary translation invariant interaction potential $v(\mathbf{r})$ between the height variables on the torus. It leads to pairs $(v,\widetilde{v})$ of mutually dual potentials and to a temperature inversion according to $\widetildeβ=π^2/β$. When $v(\mathbf{r})$ is isotropic, duality renders an anisotropic $\widetilde{v}$. This is the case, in particular, for the potential that is dual to an isotropic nearest-neighbor potential. In the thermodynamic limit this dual potential is shown to decay with distance according to an inverse square law with a quadrupolar angular dependence. There is a single pair of self-dual potentials $v^\star=\widetilde{v^\star}$. At the self-dual temperature $β^\star=\widetilde{β^\star}=π$ the height-height correlation can be calculated explicitly; it is anisotropic and diverges logarithmically with distance.

cond-mat.stat-mech↗
arXiv2021

Iterated star-triangle transformation on inhomogeneous 2D Ising lattices

We consider infinite or periodic 2D triangular Ising lattices with arbitrary positive or negative nearest-neighbor couplings $K_i(\vec{r})$, where $\vec{r}$ and $i$ indicate the bond position and orientation, respectively. Iterative application of the star-triangle transformation to an initial lattice $\mathcal{T}(0)$ with a set of couplings $\{K_i^{(0)}(\vec{r})\}$ generates a sequence of lattices $\mathcal{T}(n)$, for $n=1,2,\ldots,$ with couplings $\{K_i^{(n)}(\vec{r})\}$. When $\mathcal{T}(0)$ includes sufficiently strongly frustrated plaquettes, complex couplings will appear. We show that, nevertheless, the variables $1/\sinh 2K_i^{(n)}\!(\vec{r})$ remain confined to the union ${\mathbb{R}} \cup \rm{i}{\mathbb{R}}$ of the real and the imaginary axis. The same holds for a lattice with free boundaries, provided we distinguish between "receding" and "advancing" boundaries, the latter having degrees of freedom that must be fixed by an appropriately chosen protocol. This study establishes a framework for future analytic and numerical work on such frustrated Ising lattices.

cond-mat.stat-mech↗
arXiv2020

On the oddness of percolation

Recently Mertens and Moore [arXiv:1909.01484v1] showed that site percolation "is odd." By this they mean that on an $M\times N$ square lattice the number of distinct site configurations that allow for vertical percolation is odd. We report here an alternative proof, based on recursive use of geometric symmetry, for both free and periodic boundary conditions.

cond-mat.stat-mech↗
arXiv2019

From prelife to life: a bio-inspired toy model

We study a one-dimensional lattice of $N$ sites each occupied by a mathematical "polymer," that is, is a binary random sequence of arbitrary length $n$, or equivalently, a rooted path of $n$ links on an infinite binary tree. The average polymer length is controlled by the monomer fugacity $z$. A pair of polymers on adjacent sites carries a weight factor $ω$ for each link on the tree that they have in common. The phase diagram in the $zω$ plane exhibits a critical line $z=z_{\rm c}(ω)$. For $z z_{\rm c}(ω)$ the equilibrium is unstable and Monte Carlo time evolution brings about a dynamical symmetry breaking which favors the evolution of a small selection of polymers to ever greater length. While of interest for its own sake, this model may also be relevant to the prelife-to-life transition that has occurred during biological evolution. We compare it to existing models of similar simplicity due to Wu and Higgs (2009, 2012) and to Chen and Nowak (2012).

cond-mat.stat-mech↗
arXiv2018

Density decay and growth of correlations in the Game of Life

We study the Game of Life as a statistical system on an $L\times L$ square lattice with periodic boundary conditions. Starting from a random initial configuration of density $ρ_{\rm in}=0.3$ we investigate the relaxation of the density as well as the growth with time of spatial correlations. The asymptotic density relaxation is exponential with a characteristic time $τ_L$ whose system size dependence follows a power law $τ_L\propto L^z$ with $z=1.66\pm 0.05$ before saturating at large system sizes to a constant $τ_\infty$. The correlation growth is characterized by a time dependent correlation length $ξ_t$ that follows a power law $ξ_t\propto t^{1/z^\prime}$ with $z^\prime$ close to $z$ before saturating at large times to a constant $ξ_\infty$. We discuss the difficulty of determining the correlation length $ξ_\infty$ in the final "quiescent" state of the system. The decay time $t_{\rm q}$ towards the quiescent state is a random variable, we present simulational evidence as well as a heuristic argument indicating that for large $L$ its distribution peaks at a value $t_{\rm q}^*(L) \simeq 2τ_\infty\log L$.

cond-mat.stat-mech↗
arXiv2017

Mixed-strategy Nash equilibrium for a discontinuous symmetric $N$-player game

We consider a game in which each player must find a compromise between more daring strategies that carry a high risk for him to be eliminated, and more cautious ones that, however, reduce his final score. For two symmetric players this game was originally formulated in 1961 by Dresher, who modeled a duel between two opponents. The game has also been of interest in the description of athletic competitions. We extend here the two-player game to an arbitrary number $N$ of symmetric players. We show that there is a mixed-strategy Nash equilibrium and find its exact analytic expression, which we analyze in particular in the limit of large $N$, where mean-field behavior occurs. The original game with $N=2$ arises as a singular limit of the general case.

math.OC↗
arXiv2017

Glauber's Ising chain between two thermostats

We consider a one-dimensional Ising model each of whose $N$ spins is in contact with two thermostats of distinct temperatures $T_1$ and $T_2$. Under Glauber dynamics the stationary state happens to coincide with the equilibrium state at an effective intermediate temperature $T(T_1,T_2)$. The system nevertheless carries a nontrivial energy current between the thermostats. By means of the fermionization technique, for a chain initially in equilibrium at an arbitrary temperature $T_0$ we calculate the Fourier transform of the probability $P({\cal Q};τ)$ for the time-integrated energy current ${\cal Q}$ during a finite time interval $τ$. In the long time limit we determine the corresponding generating function for the cumulants per site and unit of time $\langle{\cal Q}^n\rangle_{\rm c}/(Nτ)$ and explicitly give those with $n=1,2,3,4.$ We exhibit various phenomena in specific regimes: kinetic mean-field effects when one thermostat flips any spin less often than the other one, as well as dissipation towards a thermostat at zero temperature. Moreover, when the system size $N$ goes to infinity while the effective temperature $T$ vanishes, the cumulants of ${\cal Q}$ per unit of time grow linearly with $N$ and are equal to those of a random walk process. In two adequate scaling regimes involving $T$ and $N$ we exhibit the dependence of the first correction upon the ratio of the spin-spin correlation length $ξ(T)$ and the size $N$.

cond-mat.stat-mech↗
arXiv2016

Imaginary noise and parity conservation in the reaction A+A <--> 0

The master equation for the reversible reaction A+A <--> 0 is considered in Poisson representation, where it is equivalent to a Langevin equation with imaginary noise for a complex stochastic variable ϕ. Such Langevin equations appear quite generally in field-theoretic treatments of reaction-diffusion problems. For this example we study the probability flow in the complex ϕplane both analytically and by simulation. We show that this flow has various curious features that must be expected to occur similarly in other Langevin equations associated with reaction-diffusion problems.

cond-mat.stat-mech↗
arXiv2016

Exact asymptotic statistics of the n-edged face in a 3D Poisson-Voronoi tessellation

We consider the 3D Poisson-Voronoi tessellation. We investigate the joint probability distribution pi_n(L) for an arbitrarily selected cell face to be n-edged and for the distance between the seeds of its adjacent cells to be equal to 2L. We derive an exact expression for this quantity, valid in the limit n->infty with n^{1/6}L fixed. The leading order correction term is determined. Good agreement with earlier Monte Carlo data is obtained. The cell face is surrounded by a three-dimensional excluded domain that is the union of n balls; it is pumpkin-shaped and analogous to the flower of the 2D Voronoi cell. For n->infty this domain tends towards a torus of equal major and minor radii. The radii scale as n^{1/3}, in agreement with earlier heuristic work. We achieve a detailed understanding of several other statistical properties of the n-edged cell face.

cond-mat.stat-mech↗
arXiv2014

Many-faced cells and many-edged faces in 3D Poisson-Voronoi tessellations

Motivated by recent new Monte Carlo data we investigate a heuristic asymptotic theory that applies to n-faced 3D Poisson-Voronoi cells in the limit of large n. We show how this theory may be extended to n-edged cell faces. It predicts the leading order large-n behavior of the average volume and surface area of the n-faced cell, and of the average area and perimeter of the n-edged face. Such a face is shown to be surrounded by a toroidal region of volume n/lambda (with lambda the seed density) that is void of seeds. Two neighboring cells sharing an n-edged face are found to have their seeds at a typical distance that scales as n^{-1/6} and whose probability law we determine. We present a new data set of 4*10^9 Monte Carlo generated 3D Poisson-Voronoi cells, larger than any before. Full compatibility is found between the Monte Carlo data and the theory. Deviations from the asymptotic predictions are explained in terms of subleading corrections whose powers in n we estimate from the data.

cond-mat.stat-mech↗
arXiv2014

A parity breaking Ising chain Hamiltonian as a Brownian motor

We consider the translationally invariant but parity (left-right symmetry) breaking Ising chain Hamiltonian \begin{equation} {\cal H} = -U_2\sum_{k} s_{k}s_{k+1} - U_3\sum_{k} s_{k}s_{k+1}s_{k+3} \nonumber \end{equation} and let this system evolve by Kawasaki spin exchange dynamics. Monte Carlo simulations show that perturbations forcing this system off equilibrium make it act as a Brownian molecular motor which, in the lattice gas interpretation, transports particles along the chain. We determine the particle current under various different circumstances, in particular as a function of the ratio $U_3/U_2$ and of the conserved magnetization $M=\sum_k s_k$. The symmetry of the $U_3$ term in the Hamiltonian is discussed

cond-mat.stat-mech↗
arXiv2014

Stripe formation instability in crossing traffic flows

At the intersection of two unidirectional traffic flows a stripe formation instability is known to occur. In this paper we consider coupled time evolution equations for the densities of the two flows in their intersection area. We show analytically how the instability arises from the randomness of the traffic entering the area. The Green function of the linearized equations is shown to form a Gaussian wave packet whose oscillations correspond to the stripes. Explicit formulas are obtained for various characteristic quantities in terms of the traffic density and comparison is made with the much simpler calculation on a torus and with numerical solution of the evolution equations.

cond-mat.stat-mech↗
arXiv2014

Large-n conditional facedness m_n of 3D Poisson-Voronoi cells

We consider the three-dimensional Poisson-Voronoi tessellation and study the average facedness m_n of a cell known to neighbor an n-faced cell. Whereas Aboav's law states that m_n=A+B/n, theoretical arguments indicate an asymptotic expansion m_n = 8 + k_1 n^{-1/6} +.... Recent new Monte Carlo data due to Lazar et al., based on a very large data set, now clearly rule out Aboav's law. In this work we determine the numerical value of k_1 and compare the expansion to the Monte Carlo data. The calculation of k_1 involves an auxiliary planar cellular structure composed of circular arcs, that we will call the Poisson-Moebius diagram. It is a special case of more general Moebius diagrams (or multiplicatively weighted power diagrams) and is of interest for its own sake. We obtain exact results for the total edge length per unit area, which is a prerequisite for the coefficient k_1, and a few other quantities in this diagram.

cond-mat.stat-mech↗
arXiv2013

Exact domain wall theory for deterministic TASEP with parallel update

Domain wall theory (DWT) has proved to be a powerful tool for the analysis of one-dimensional transport processes. A simple version of it was found very accurate for the Totally Asymmetric Simple Exclusion Process (TASEP) with random sequential update. However, a general implementation of DWT is still missing in the case of updates with less fluctuations, which are often more relevant for applications. Here we develop an exact DWT for TASEP with parallel update and deterministic (p=1) bulk motion. Remarkably, the dynamics of this system can be described by the motion of a domain wall not only on the coarse-grained level but also exactly on the microscopic scale for arbitrary system size. All properties of this TASEP, time-dependent and stationary, are shown to follow from the solution of a bivariate master equation whose variables are not only the position but also the velocity of the domain wall. In the continuum limit this exactly soluble model then allows us to perform a first principle derivation of a Fokker-Planck equation for the position of the wall. The diffusion constant appearing in this equation differs from the one obtained with the traditional `simple' DWT.

cond-mat.stat-mech↗
arXiv2013

Crossing pedestrian traffic flows,diagonal stripe pattern, and chevron effect

We study two perpendicular intersecting flows of pedestrians. The latter are represented either by moving hard core particles of two types, eastbound ($\symbp$) and northbound ($\symbm$), or by two density fields, $\rhop_t(\brr)$ and $\rhom_t(\brr)$. Each flow takes place on a lattice strip of width $M$ so that the intersection is an $M\times M$ square. We investigate the spontaneous formation, observed experimentally and in simulations, of a diagonal pattern of stripes in which alternatingly one of the two particle types dominates. By a linear stability analysis of the field equations we show how this pattern formation comes about. We focus on the observation, reported recently, that the striped pattern actually consists of chevrons rather than straight lines. We demonstrate that this `chevron effect' occurs both in particle simulations with various different update schemes and in field simulations. We quantify the effect in terms of the chevron angle $Δθ_0$ and determine its dependency on the parameters governing the boundary conditions.

cond-mat.stat-mech↗
arXiv2012

A multi-lane TASEP model for crossing pedestrian traffic flows

A one-way {\em street} of width M is modeled as a set of M parallel one-dimensional TASEPs. The intersection of two perpendicular streets is a square lattice of size M times M. We consider hard core particles entering each street with an injection probability α. On the intersection square the hard core exclusion creates a many-body problem of strongly interacting TASEPs and we study the collective dynamics that arises. We construct an efficient algorithm that allows for the simulation of streets of infinite length, which have sharply defined critical jamming points. The algorithm employs the `frozen shuffle update', in which the randomly arriving particles have fully deterministic bulk dynamics. High precision simulations for street widths up to M=24 show that when αincreases, there occur jamming transitions at a sequence of M critical values \alphaM,M < \alphaM,M-1 < ... < \alphaM,1. As M grows, the principal transition point \alphaM,M decreases roughly as \sim 1/(log M) in the range of M values studied. We show that a suitable order parameter is provided by a reflection coefficient associated with the particle current in each TASEP.

cond-mat.stat-mech↗