Rowlinson Model Research Articles

Non-hyperuniformity of Gibbs point processes with short-range interactions

Abstract We investigate the hyperuniformity of marked Gibbs point processes that have weak dependencies among distant points whilst the interactions of close points are kept arbitrary. Various stability and range assumptions are imposed on the Papangelou intensity in order to prove that the resulting point process is not hyperuniform. The scope of our results covers many frequently used models, including Gibbs point processes with a superstable, lower-regular, integrable pair potential, as well as the Widom–Rowlinson model with random radii and Gibbs point processes with interactions based on Voronoi tessellations and nearest-neighbour graphs.

Dynamical Gibbs–non-Gibbs transitions in Widom–Rowlinson models on trees

We consider the soft-core Widom-Rowlinson model for particles with spins and holes, on a Cayley tree of order $d$ (which has $d + 1$ nearest neighbours), depending on repulsion strength $\beta$ between particles of different signs and on an activity parameter $\lambda$ for particles. We analyse Gibbsian properties of the time-evolved intermediate Gibbs measure of the static model, under a spin-flip time evolution, in a regime of large repulsion strength $\beta$. We first show that there is a dynamical transition, in which the measure becomes non-Gibbsian at large times, independently of the particle activity, for any $d \geq 2$. In our second and main result, we also show that for large $\beta$ and at large times, the measure of the set of bad configurations (discontinuity points) changes from zero to one as the particle activity $\lambda$ increases, assuming that $d \geq 4$. Our proof relies on a general zero-one law for bad configurations on the tree, and the introduction of a set of uniformly bad configurations given in terms of subtree percolation, which we show to become typical at high particle activity.

Sharp phase transition for the continuum Widom–Rowlinson model

The Widom–Rowlinson model (or the Area-interaction model) is a Gibbs point process in Rd with the formal Hamiltonian defined as the volume of ∪x∈ωB1(x), where ω is a locally finite configuration of points and B1(x) denotes the unit closed ball centred at x. The model is also tuned by two other parameters: the activity z>0 related to the intensity of the process and the inverse temperature β≥0 related to the strength of the interaction. In the present paper we investigate the phase transition of the model in the point of view of percolation theory and the liquid-gas transition. First, considering the graph connecting points with distance smaller than 2r>0, we show that for any β≥0, there exists 0 z˜ca(β,r)). These results are in the spirit of recent works using the theory of randomised tree algorithms (Probab. Theory Related Fields 173 (2019) 479–490, Ann. of Math. 189 (2019) 75–99, Duminil-Copin, Raoufi and Tassion (2018)). Secondly we study a standard liquid-gas phase transition related to the uniqueness/non-uniqueness of Gibbs states depending on the parameters z, β. Old results (Phys. Rev. Lett. 27 (1971) 1040–1041, J. Chem. Phys. 52 (1970) 1670–1684) claim that a non-uniqueness regime occurs for z=β large enough and it is conjectured that the uniqueness should hold outside such an half line (z=β≥βc>0). We solve partially this conjecture in any dimension by showing that for β large enough the non-uniqueness holds if and only if z=β. We show also that this critical value z=β corresponds to the percolation threshold z˜ca(β,r)=β for β large enough, providing a straight connection between these two notions of phase transition.

Finitary codings for spatial mixing Markov random fields

It has been shown by van den Berg and Steif (Ann. Probab. 27 (1999) 1501–1522) that the subcritical and critical Ising model on $\mathbb{Z}^{d}$ is a finitary factor of an i.i.d. process (ffiid), whereas the super-critical model is not. In fact, they showed that the latter is a general phenomenon in that a phase transition presents an obstruction for being ffiid. The question remained whether this is the only such obstruction. We make progress on this, showing that certain spatial mixing conditions (notions of weak dependence on boundary conditions, not to be confused with other notions of mixing in ergodic theory) imply ffiid. Our main result is that weak spatial mixing implies ffiid with power-law tails for the coding radius, and that strong spatial mixing implies ffiid with exponential tails for the coding radius. The weak spatial mixing condition can be relaxed to a condition which is satisfied by some critical two-dimensional models. Using a result of the author (Spinka (2018)), we deduce that strong spatial mixing also implies ffiid with stretched-exponential tails from a finite-valued i.i.d. process. We give several applications to models such as the Potts model, proper colorings, the hard-core model, the Widom–Rowlinson model and the beach model. For instance, for the ferromagnetic $q$-state Potts model on $\mathbb{Z}^{d}$ at inverse temperature $\beta $, we show that it is ffiid with exponential tails if $\beta $ is sufficiently small, it is ffiid if $\beta <\beta _{c}(q,d)$, it is not ffiid if $\beta >\beta_{c}(q,d)$ and, when $d=2$ and $\beta =\beta _{c}(q,d)$, it is ffiid if and only if $q\le 4$.

Gibbsian Representation for Point Processes via Hyperedge Potentials

We consider marked point processes on the d-dimensional Euclidean space, defined in terms of a quasilocal specification based on marked Poisson point processes. We construct absolutely summable Hamiltonians in terms of hyperedge potentials in the sense of Georgii et al. (Probab Theory Relat Fields 153(3–4):643–670, 2012), which are useful in models of stochastic geometry. These potentials allow for weak non-localities and are a natural generalization of the usual physical multi-body potentials, which are strictly local. Our proof relies on regrouping arguments, which use the possibility of controlled non-localities in the class of hyperedge potentials. As an illustration, we also provide such representations for the Widom–Rowlinson model under independent spin-flip time evolution. With this work, we aim to draw a link between the abstract theory of point processes in infinite volume, the study of measures under transformations and statistical mechanics of systems of point particles.

Hard-core and soft-core Widom–Rowlinson models on Cayley trees

We consider both hard-core and soft-core Widom–Rowlinson models with spin values on a Cayley tree of order and we are interested in the Gibbs measures of the models. The models depend on three parameters: the order k of the tree, describing the strength of the (ferromagnetic or antiferromagnetic) interaction, and describing the intensity for particles. The hard-core Widom–Rowlinson model corresponds to the case .For the binary tree k = 2, and for k = 3 we prove that the ferromagnetic model has either one or three splitting Gibbs measures (tree-automorphism invariant Gibbs measures (TISGMs) which are tree-indexed Markov chains). We also give the exact form of the corresponding critical curves in parameter space. For higher values of k we give an explicit sufficient bound ensuring non-uniqueness which we conjecture to be the exact curve. Moreover, for the antiferromagnetic model we explicitly give two critical curves , , and prove that on these curves there are exactly two TISGMs; between these curves there are exactly three TISGMs; otherwise there exists a unique TISGM. Some periodic and non-periodic SGMs are also constructed in the ferromagnetic model.

The Widom-Rowlinson model on the Delaunay graph

We establish phase transitions for continuum Delaunay multi-type particle systems (continuum Potts or Widom-Rowlinson models) with a repulsive interaction between particles of different types. Our interaction potential depends solely on the length of the Delaunay edges. We show that a phase transition occurs for sufficiently large activities and for sufficiently large potential parameter proving an old conjecture of Lebowitz and Lieb extended to the Delaunay structure. Our approach involves a Delaunay random-cluster representation analogous to the Fortuin-Kasteleyn representation of the Potts model. The phase transition manifests itself in the mixed site-bond percolation of the corresponding random-cluster model. Our proofs rely mainly on geometric properties of Delaunay tessellations in $\mathbb{R} ^{2} $ and on recent studies [DDG12] of Gibbs measures for geometry-dependent interactions. The main tool is a uniform bound on the number of connected components in the Delaunay graph which provides a novel approach to Delaunay Widom Rowlinson models based on purely geometric arguments. The interaction potential ensures that shorter Delaunay edges are more likely to be open and thus offsets the possibility of having an unbounded number of connected components.

Phase Transition for Continuum Widom–Rowlinson Model with Random Radii

In this paper we study the phase transition of continuum Widom-Rowlinson measures in $\mathbb{R}^d$ with $q$ types of particles and random radii. Each particle $x_i$ of type $i$ is marked by a random radius $r_i$ distributed by a probability measure $Q_i$ on $\mathbb{R}^+$. The particles of same type do not interact each other whereas particles $x_i$ and $x_j$ with different type $i \neq j$ interact via an exclusion hardcore interaction forcing $r_i+r_j$ to be smaller than $|x_i-x_j|$. In the integrable case (i.e. $\int r^d Q_i(dr)<+\infty$, $1\le i\le q$), we show that the Widom-Rowlinson measures exhibit a standard phase transition providing uniqueness, when the activity is small, and co-existence of $q$ ordered phases, when the activity is large. In the non-integrable case (i.e. $\int r^d Q_i(dr)=+\infty$, $1\le i \le q$), we show another type of phase transition. We prove, when the activity is small, the existence of at least $q+1$ extremal phases and we conjecture that, when the activity is large, only the $q$ ordered phases subsist. We prove a weak version of this conjecture by showing that the symmetric Widom-Rowlinson measure with free boundary condition is a mixing of the $q$ ordered phases if and only if the activity is large.

A phase transition in a Widom–Rowlinson model with Curie–Weiss interaction

An analog of the continuum Widom–Rowlinson model is introduced and studied. Its two-component version is a gas of point particles of types 0 and 1 placed in , in which like particles do not interact, whereas unlike particles contained in a vessel of volume repel each other with intensity . The one-component version is a gas of particles with multi-particle interactions of Curie–Weiss type. Its thermodynamic behavior is obtained by integrating out the coordinates of one of the components of the two-component version. In the grand canonical setting, a rigorous theory of phase transitions in this model is developed and discussed. In particular, for both versions, thermodynamic phases and phase diagrams are explicitly constructed and the equations of state are obtained and analyzed.

Percolation results for the continuum random cluster model

Abstract The continuum random cluster model is a Gibbs modification of the standard Boolean model with intensity z &amp;gt; 0 and law of radii Q. The formal unnormalised density is given by q N cc , where q is a fixed parameter and N cc is the number of connected components in the random structure. We prove for a large class of parameters that percolation occurs for large enough z and does not occur for small enough z. We provide an application to the phase transition of the Widom–Rowlinson model with random radii. Our main tools are stochastic domination properties, a detailed study of the interaction of the model, and a Fortuin–Kasteleyn representation.

A Curie–Weiss theory of the continuum Widom–Rowlinson model

A version of the continuum Widom–Rowlinson model is introduced and studied. It is a two-component gas of point particles placed in Rd in which like particles do not interact and unlike particles contained in a given vessel of volume V repel each other with intensity a/V. This model is thermodynamically equivalent to a one-component gas with multi-particle interaction. For both models, a rigorous theory of a phase transition is presented and the ways of its construction in the framework of the grand canonical formalism are outlined.

The Widom–Rowlinson model under spin flip: Immediate loss and sharp recovery of quasilocality

We consider the continuum Widom–Rowlinson model under independent spin-flip dynamics and investigate whether and when the time-evolved point process has an (almost) quasilocal specification (Gibbs-property of the time-evolved measure). Our study provides a first analysis of a Gibbs–non-Gibbs transition for point particles in Euclidean space. We find a picture of loss and recovery, in which even more regularity is lost faster than it is for time-evolved spin models on lattices. We show immediate loss of quasilocality in the percolation regime, with full measure of discontinuity points for any specification. For the color-asymmetric percolating model, there is a transition from this non-almost-sure quasilocal regime back to an everywhere Gibbsian regime. At the sharp reentrance time $t_{G}>0$, the model is a.s. quasilocal. For the color-symmetric model, there is no reentrance. On the constructive side, for all $t>t_{G}$, we provide everywhere quasilocal specifications for the time-evolved measures and give precise exponential estimates on the influence of boundary condition.

The Widom–Rowlinson model, the hard-core model and the extremality of the complete graph

Let HWR be the path on 3 vertices with a loop at each vertex. D. Galvin (Galvin, 2013, 2014) conjectured, and E. Cohen, W. Perkins and P. Tetali (Cohen et al., in press) proved that for any d-regular simple graph G on n vertices we have hom(G,HWR)≤hom(Kd+1,HWR)n/(d+1). In this paper we give a short proof of this theorem together with the proof of a conjecture of Cohen, Perkins and Tetali (Cohen et al., in press). Our main tool is a simple bijection between the Widom–Rowlinson model and the hard-core model on another graph. We also give a large class of graphs H for which we have hom(G,H)≤hom(Kd+1,H)n/(d+1). In particular, we show that the above inequality holds if H is a path or a cycle of even length at least 6 with loops at every vertex.

On the Widom–Rowlinson Occupancy Fraction in Regular Graphs

We consider the Widom–Rowlinson model of two types of interacting particles on d-regular graphs. We prove a tight upper bound on the occupancy fraction, the expected fraction of vertices occupied by a particle under a random configuration from the model. The upper bound is achieved uniquely by unions of complete graphs on d + 1 vertices, Kd+1. As a corollary we find that Kd+1 also maximizes the normalized partition function of the Widom–Rowlinson model over the class of d-regular graphs. A special case of this shows that the normalized number of homomorphisms from any d-regular graph G to the graph HWR, a path on three vertices with a loop on each vertex, is maximized by Kd+1. This proves a conjecture of Galvin.

A Classical WR Model with $$q$$ q Particle Types

A version of the Widom–Rowlinson model is considered, where particles of $$q$$ types coexist, subject to pairwise hard-core exclusions. For $$q\le 4$$ , 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.

Dynamical Widom–Rowlinson Model and Its Mesoscopic Limit

We consider the non-equilibrium dynamics for the Widom–Rowlinson model (without hard-core) in the continuum. The Lebowitz–Penrose-type scaling of the dynamics is studied and the system of the corresponding kinetic equations is derived. In the space-homogeneous case, the equilibrium points of this system are described. Their structure corresponds to the dynamical phase transition in the model. The bifurcation of the system is shown.

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.

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

The paper extends earlier results from Suhov and Kelbert [“FK-DLR states of a quantum Bose-gas with a hardcore interaction,” arXiv:1304.0782] and Suhov et al. [“Shift-invariance for FK-DLR states of a 2D quantum Bose-gas,” arXiv:1304.4177] 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 Suhov and Kelbert and Suhov et al., we establish that, for the values of fugacity z ∈ (0, 1) and inverse temperature β &amp;gt; 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 Suhov and Kelbert and Suhov et al.) is translation-invariant.

ONE-DIMENSIONAL LONG RANGE WIDOM–ROWLINSON MODEL WITH PERIODIC PARTICLE ACTIVITIES

In this paper, we consider a one-dimensional long range Widom–Rowlinson model when particle activity parameters are periodic and biased. We show that if the interaction is sufficiently large versus particle activities then the model does not exhibit a phase transition at low temperatures.

Extremal graphs for homomorphisms

The study of graph homomorphisms has a long and distinguished history, with applications in many areas of graph theory. There has been recent interest in counting homomorphisms, and in particular on the question of finding upper bounds for the number of homomorphisms from a graph G into a fixed image graph H. We introduce our techniques by proving that the lex graph has the largest number of homomorphisms into K2 with one looped vertex (or equivalently, the largest number of independent sets) among graphs with fixed number of vertices and edges. Our main result is the solution to the extremal problem for the number of homomorphisms into P, the completely looped path of length 2 (known as the Widom–Rowlinson model in statistical physics). We show that there are extremal graphs that are threshold, give explicitly a list of five threshold graphs from which any threshold extremal graph must come, and show that each of these “potentially extremal” threshold graphs is in fact extremal for some number of edges. Copyright © 2011 Wiley Periodicals, Inc. J Graph Theory 67: 261–284, 2011