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L. Chayes

Publications and source records attributed to L. Chayes.

17 recordsLinked to original sources
arXiv2016

Transport and Equilibrium in Non-Conservative Systems

We study, in finite volume, a grand canonical version of the McKean-Vlasov equation where the total particle content is allowed to vary. The dynamics is anticipated to minimize an appropriate grand canonical free energy; we make this notion precise by introducing a metric on a set of positive Borel measures without pre-prescribed mass and demonstrating that the dynamics is a gradient flow with respect to this metric. Moreover, we develop a JKO-scheme suitable for these problems. The latter ideas have general applicability to a class of second order non-conservative problems. For this particular system we prove, using the JKO-scheme, that (under certain assumptions) convergence to the uniform stationary state is exponential with a rate which is independent of the volume. By contrast, in related conservative systems, decay rates scale - at best - with the square of the characteristic length of the system. This suggests that a grand canonical approach may be useful for both theoretical and computational study of large scale systems.

math.AP↗
arXiv2013

On the Rate of Convergence for Critical Crossing Probabilities

For the site percolation model on the triangular lattice and certain generalizations for which Cardy's Formula has been established we acquire a power law estimate for the \emph{rate} of convergence of the crossing probabilities to Cardy's Formula.

math-ph↗
arXiv2011

Hamiltonian ODE's on a Space of Deficient Measures

We continue the study (initiated in [1]) of Borel measures whose time evolution is provided by an interacting Hamiltonian structure. Here, the principal focus is the development and advancement of deficency in the measure caused by displacement of mass to infinity in finite time. We introduce - and study in its own right - a regularization scheme based on a dissipative mechanism which naturally degrades mass according to distance traveled (in phase space). Our principal results are obtained based on some dynamical considerations in the form of a condition which forbids mass to return from infinity.

math.AP↗
arXiv2010

Conformal Invariance for Certain Models of the Bond-Triangular Type

Following the approach outlined in [18], convergence to SLE6 of the Exploration Processes for the correlated bond-triangular type models studied in [7] is established. This puts the said models in the same universality class as the standard site percolation model on the triangular lattice [19]. The result is proven for all domains with boundary (upper) Minkowski dimension less than two. Moreover, the proof of convergence applies in the general context of critical 2D percolation models, under the stipulation that Cardy's Formula can be established.

math-ph↗
arXiv2010

On Convergence to SLE$_6$ I: Conformal Invariance for Certain Models of the Bond-Triangular Type

Following the approach outlined in [26], convergence to SLE$_6$ of the Exploration Processes for the correlated bond-triangular type models studied in [11] is established. This puts the said models in the same universality class as the standard site percolation model on the triangular lattice [27]. In the context of these models, the result is proven for all domains with boundary Minkowski dimension less than two. Moreover, the proof of convergence applies in the context of general critical 2D percolation models and for general domains, under the stipulation that Cardy's Formula can be established for domains in this generality.

math-ph↗
arXiv2007

Cardy's Formula for Certain Models of the Bond-Triangular Type

We introduce and study a family of 2D percolation systems which are based on the bond percolation model of the triangular lattice. The system under study has local correlations, however, bonds separated by a few lattice spacings act independently of one another. By avoiding explicit use of microscopic paths, it is first established that the model possesses the typical attributes which are indicative of critical behavior in 2D percolation problems. Subsequently, the so called Cardy-Carleson functions are demonstrated to satisfy, in the continuum limit, Cardy's formula for crossing probabilities. This extends the results of S. Smirnov to a non-trivial class of critical 2D percolation systems.

math-ph↗
arXiv2007

Percolation Phenomena in Low and High Density Systems

We consider the 2D quenched--disordered $q$--state Potts ferromagnets and show that at self--dual points any amalgamation of $q-1$ species will fail to percolate despite an overall (high) density of $1-q^{-1}$. Further, in the dilute bond version of these systems, if the system is just above threshold, then throughout the low temperature phase there is percolation of a single species despite a correspondingly small density. Finally, we demonstrate both phenomena in a single model by considering a ``perturbation'' of the dilute model that has a self--dual point. We also demonstrate that these phenomena occur, by a similar mechanism, in a simple coloring model invented by O. Häggström.

cond-mat.stat-mech↗
arXiv2005

Random Cluster Models on the Triangular Lattice

We study percolation and the random cluster model on the triangular lattice with 3-body interactions. Starting with percolation, we generalize the star--triangle transformation: We introduce a new parameter (the 3-body term) and identify configurations on the triangles solely by their connectivity. In this new setup, necessary and sufficient conditions are found for positive correlations and this is used to establish regions of percolation and non-percolation. Next we apply this set of ideas to the $q>1$ random cluster model: We derive duality relations for the suitable random cluster measures, prove necessary and sufficient conditions for them to have positive correlations, and finally prove some rigorous theorems concerning phase transitions.

cond-mat.stat-mech↗
arXiv2004

Vertex Models and Random Labyrinths: Phase Diagrams for Ice-type Vertex Models

We propose a simple geometric recipe for constructing phase diagrams for a general class of vertex models obeying the ice rule. The disordered phase maps onto the intersecting loop model which is interesting in its own right and is related to several other statistical mechanical models. This mapping is also useful in understanding some ordered phases of these vertex models as they correspond to the polymer loop models with cross-links in their vulcanised phase.

cond-mat.stat-mech↗
arXiv2001

Invaded cluster simulations of the XY model in two and three dimensions

The invaded cluster algorithm is used to study the XY model in two and three dimensions up to sizes 2000^2 and 120^3 respectively. A soft spin O(2) model, in the same universality class as the 3D XY model, is also studied. The static critical properties of the model and the dynamical properties of the algorithm are reported. The results are K_c=0.45412(2) for the 3D XY model and eta=0.037(2) for the 3D XY universality class. For the 2D XY model the results are K_c=1.120(1) and eta=0.251(5). The invaded cluster algorithm does not show any critical slowing for the magnetization or critical temperature estimator for the 2D or 3D XY models.

cond-mat.stat-mech↗
arXiv1999

Lebowitz Inequalities for Ashkin-Teller Systems

We consider the Ashkin-Teller model with negative four-spin coupling but still in the region where the ground state is ferromagnetic. We establish the standard Lebowitz inequality as well as the extension that is necessary to prove a divergent susceptibility.

cond-mat.stat-mech↗
arXiv1999

Intersecting Loop Models on Z^D: Rigorous Results

We consider a general class of (intersecting) loop models in D dimensions, including those related to high-temperature expansions of well-known spin models. We find that the loop models exhibit some interesting features - often in the ``unphysical'' region of parameter space where all connection with the original spin Hamiltonian is apparently lost. For a particular n=2, D=2 model, we establish the existence of a phase transition, possibly associated with divergent loops. However, for n >> 1 and arbitrary D there is no phase transition marked by the appearance of large loops. Furthermore, at least for D=2 (and n large) we find a phase transition characterised by broken translational symmetry.

cond-mat.stat-mech↗
arXiv1998

Graphical Representations for Ising Systems in External Fields

A graphical representation based on duplication is developed that is suitable for the study of Ising systems in external fields. Two independent replicas of the Ising system in the same field are treated as a single four-state (Ashkin-Teller) model. Bonds in the graphical representation connect the Ashkin-Teller spins. For ferromagnetic systems it is proved that ordering is characterized by percolation in this representation. The representation leads immediately to cluster algorithms; some applications along these lines are discussed.

cond-mat.stat-mech↗
arXiv1998

The Mean Field Theory for Percolation Models of the Ising Type

The $q=2$ random cluster model is studied in the context of two mean field models: The Bethe lattice and the complete graph. For these systems, the critical exponents that are defined in terms of finite clusters have some anomalous values as the critical point is approached from the high density side which vindicates the results of earlier studies. In particular, the exponent $\tilde γ^\prime$ which characterises the divergence of the average size of finite clusters is 1/2 and $\tildeν^\prime$, the exponent associated with the length scale of finite clusters is 1/4. The full collection of exponents indicates an upper critical dimension of 6. The standard mean field exponents of the Ising system are also present in this model ($ν^\prime = 1/2$, $γ^\prime = 1$) which implies, in particular, the presence of two diverging length scales. Furthermore, the finite cluster exponents are stable to the addition of disorder which, near the upper critical dimension, may have interesting implications concerning the generality of the disordered system/correlation length bounds.

cond-mat.stat-mech↗
arXiv1995

Avoided Critical Behavior in a Uniformly Frustrated System

We study the effects of weak long-ranged antiferromagnetic interactions of strength $Q$ on a spin model with predominant short-ranged ferromagnetic interactions. In three dimensions, this model exhibits an avoided critical point in the sense that the critical temperature $T_c(Q=0)$ is strictly greater than $\lim_{Q\to 0} T_c(Q)$. The behavior of this system at temperatures less than $T_c(Q=0)$ is controlled by the proximity to the avoided critical point. We also quantize the model in a novel way to study the interplay between charge-density wave and superconducting order.

cond-mat↗