Metastable behavior for bootstrap percolation on regular trees (2009)
Biskup, Marek, Schonmann, Roberto H.
We examine bootstrap percolation on a regular (b+1)-ary tree with initial law given by Bernoulli(p). The sites are updated according to the usual rule: a vacant site becomes occupied if it has at...
Colligative properties of solutions: I. Fixed concentrations, previous paper (2008)
Kenneth S. Alex, Marek Biskup, Lincoln Chayes
Using the formalism of rigorous statistical mechanics, we study the phenomena of phase separation and freezing-point depression upon freezing of solutions. Specifically, we devise an Ising-based...
LARGE-DEVIATIONS/THERMODYNAMIC APPROACH TO PERCOLATION ON THE COMPLETE GRAPH (2008)
Marek Biskup, Lincoln Chayes, S. Alex Smith
ABSTRACT. We present a large-deviations/thermodynamic approach to the classic problem of percolation on the complete graph. Specifically, we determine the large-deviation rate function for the...
Marek Biskup, Lincoln Chayes, Roman Kotecky
We study equilibrium droplets in two-phase systems at parameter values corresponding to phase coexistence. Specifically, we give a self-contained microscopic
LARGE-DEVIATIONS/THERMODYNAMIC APPROACH TO PERCOLATION ON THE COMPLETE GRAPH (2008)
Marek Biskup, Lincoln Chayes, S. Alex Smith
ABSTRACT. We present a large-deviations/thermodynamic approach to the classic problem of percolation on the complete graph. Specifically, we determine the large-deviation rate function for the...
GRAPH DIAMETER IN LONG-RANGE PERCOLATION 1.1 Overview. (2008)
Abstract: We study the asymptotic growth of the diameter of the graph obtained by adding sparse “long ” edges to a square box in Z d. We focus on the cases when an edge between x and y is added...
Abstract: We study the formation/dissolution of equilibrium droplets in finite systems at parameters corresponding to phase coexistence. Specifically, we consider the 2D Ising model in volumes of...
Abstract: We consider the parabolic Anderson problem @ t u = \Deltau + u on (0; 1) \Theta Z d with random i.i.d. potential = ((z)) z2Z d and the initial condition u(0; \Delta) j 1. Our main...
Functional CLT for random walk among bounded random conductances (2007)
Biskup, Marek; UCLA; Biskup@math.ucla.edu, Prescott, Timothy M; UCLA; Tmpresco@math.ucla.edu
We consider the nearest-neighbor simple random walk on Zd, d≥2, driven by a field of i.i.d. random nearest-neighbor conductances ωxy∈[0,1]. Apart from the requirement that the bonds...
Scaling limit for a class of gradient fields with non-convex potentials (2007)
We consider gradient fields $(\phi_x\colon x\in\Z^d)$ whose law takes the Gibbs-Boltzmann form $Z^{-1}\exp\{-\sum_{< x,y>}V(\phi_y-\phi_x)\}$ where the sum runs over nearest neighbors. We assume that...
Functional CLT for random walk among bounded random conductances (2007)
Biskup, Marek, Prescott, Timothy M.
We consider the nearest-neighbor simple random walk on $\Z^d$, $d\ge2$, driven by a field of i.i.d. random nearest-neighbor conductances $\omega_{xy}\in[0,1]$. Apart from the requirement that the...
Anomalous heat-kernel decay for random walk among bounded random conductances (2006)
Berger, Noam, Biskup, Marek, Hoffman, Christopher E., Kozma, Gady
We consider the nearest-neighbor simple random walk on $\Z^d$, $d\ge2$, driven by a field of bounded random conductances $\omega_{xy}\in[0,1]$. The conductance law is i.i.d. subject to the condition...
Reflection positivity and phase transitions in lattice spin models (2006)
Reflection positivity (RP) is a property of Gibbs measures exhibited by a class of lattice spin systems that includes the Ising, Potts and Heisenberg models. The RP property is useful because of its...
On the absence of ferromagnetism in typical 2D ferromagnets (2006)
Biskup, Marek, Chayes, Lincoln, Kivelson, Steven A.
We consider the Ising systems in $d$ dimensions with nearest-neighbor ferromagnetic interactions and long-range repulsive (antiferromagnetic) interactions which decay with a power, $s$, of the...
Marek Biskup, Lincoln Chayes, Nicholas Crawford
We consider a class of spin systems on Z d with vector valued spins (Sx) that interact via the pair-potentials Jx,y Sx · Sy. The interactions are generally spread-out in the sense that the Jx,y’s...
Functional CLT for random walk among bounded conductances (2006)
ABSTRACT. We consider the nearest-neighbor simple random walk on Z d, d ≥ 2, driven by a field of i.i.d. random nearest-neighbor conductances ωxy ∈ [0, 1]. Apart from the requirement that the...
Phase coexistence of gradient Gibbs states (2005)
We consider the (scalar) gradient fields $\eta=(\eta_b)$--with $b$ denoting the nearest-neighbor edges in $\Z^2$--that are distributed according to the Gibbs measure proportional to $\texte^{-\beta...
Quantum spin systems at positive temperature (2005)
Biskup, Marek, Chayes, Lincoln, Starr, Shannon
We develop a novel approach to phase transitions in quantum spin models based on a relation to their classical counterparts. Explicitly, we show that whenever chessboard estimates can be used to...
Large-deviations/thermodynamic approach to percolation on the complete graph (2005)
Biskup, Marek, Chayes, Lincoln, Smith, S. Alex
We present a large-deviations/thermodynamic approach to the classic problem of percolation on the complete graph. Specifically, we determine the large-deviation rate function for the probability that...
Forbidden gap argument for phase transitions proved by means of chessboard estimates (2005)
Chessboard estimates are one of the standard tools for proving phase coexistence in spin systems of physical interest. In this note we show that the method not only produces a point in the phase...
Quenched invariance principle for simple random walk on percolation clusters (2005)
We consider the simple random walk on the (unique) infinite cluster of super-critical bond percolation in $\Z^d$ with $d\ge2$. We prove that, for almost every percolation configuration, the path...
Mean-field driven first-order phase transitions in systems with long-range interactions (2005)
Biskup, Marek, Chayes, Lincoln, Crawford, Nicholas
We consider a class of spin systems on $\Z^d$ with vector valued spins $(\bS_x)$ that interact via the pair-potentials $J_{x,y} \bS_x\cdot\bS_y$. The interactions are generally spread-out in the...
Marek Biskup, Lincoln Chayes, Shannon Starr
Abstract: We develop a novel approach to phase transitions in quantum spin models based on a relation to their classical counterparts. Explicitly, we show that whenever chessboard estimates can be...
On the scaling of the chemical distance in long-range percolation models (2004)
We consider the (unoriented) long-range percolation on ℤd in dimensions d≥1, where distinct sites x,y∈ℤd get connected with probability pxy∈[0,1]. Assuming pxy=|x−y|−s+o(1) as...
Colligative properties of solutions: I. Fixed concentrations (2004)
Alexander, Kenneth, Biskup, Marek, Chayes, Lincoln
Using the formalism of rigorous statistical mechanics, we study the phenomena of phase separation and freezing-point depression upon freezing of solutions. Specifically, we devise an Ising-based...
Colligative properties of solutions: II. Vanishing concentrations (2004)
Alexander, Kenneth, Biskup, Marek, Chayes, Lincoln
We continue our study of colligative properties of solutions initiated in math-ph/0407034. We focus on the situations where, in a system of linear size $L$, the concentration and the chemical...
Graph diameter in long-range percolation (2004)
We study the asymptotic growth of the diameter of the graph obtained by adding sparse "long" edges to a square box in $Z^{d}$. We focus on the cases when an edge between $x$ and $y$ is added with...
S.A.: Order by disorder, without order, in a two-dimensional spin system with O(2)-symmetry (2004)
Marek Biskup, Lincoln Chayes, Steven A. Kivelson
Abstract. We present a rigorous proof of an ordering transition for a two-component two-dimensional antiferromagnet with nearest and next-nearest neighbor interactions. The low-temperature phase...
Colligative Properties of Solutions: II. Vanishing Concentrations (2004)
Kenneth S. Alex, Marek Biskup, Lincoln Chayes
We continue our study of colligative properties of solutions initiated in ref. 1. We focus on the situations where, in a system of linear size L, the concentration and the chemical potential scale...
Partition function zeros at first-order phase transitions: Pirogov-Sinai theory (2003)
Biskup, Marek, Borgs, Christian, Chayes, Jennifer T., Kotecky, Roman
This paper is a continuation of our previous analysis [BBCKK] of partition functions zeros in models with first-order phase transitions and periodic boundary conditions. Here it is shown that the...
Order by disorder, without order, in a two-dimensional spin system with O(2) symmetry (2003)
Biskup, Marek, Chayes, Lincoln, Kivelson, Steven A.
We present a rigorous proof of an ordering transition for a two-component two-dimensional antiferromagnet with nearest and next-nearest neighbor interactions. The low-temperature phase contains two...
Orbital ordering in transition-metal compounds: I. The 120-degree model (2003)
Biskup, Marek, Chayes, Lincoln, Nussinov, Zohar
We study the classical version of the 120-degree model. This is an attractive nearest-neighbor system in three dimensions with XY (rotor) spins and interaction such that only a particular projection...
Orbital order in classical models of transition-metal compounds (2003)
Nussinov, Zohar, Biskup, Marek, Chayes, Lincoln, Brink, Jeroen Van Den
We study the classical 120-degree and related orbital models. These are the classical limits of quantum models which describe the interactions among orbitals of transition-metal compounds. We...
On the scaling of the chemical distance in long-range percolation models (2003)
We consider the (unoriented) long-range percolation on Z^d in dimensions d\ge1, where distinct sites x,y\in Z^d get connected with probability p_{xy}\in[0,1]. Assuming p_{xy}=|x-y|^{-s+o(1)} as...
Partition function zeros at first-order phase transitions: A general analysis (2003)
Biskup, Marek, Borgs, Christian, Chayes, Jennifer T., Kleinwaks, Logan J., Kotecky, Roman
We present a general, rigorous theory of partition function zeros for lattice spin models depending on one complex parameter. First, we formulate a set of natural assumptions which are verified for a...
Biskup, Marek, Chayes, Lincoln, Kotecky, Roman
We examine some aspects of the recent results by K. Binder. The equilibrium formation/dissolution of droplets in finite systems is discussed in the context of the canonical and the grand canonical...
A proof of the Gibbs-Thomson formula in the droplet formation regime (2003)
Biskup, Marek, Chayes, Lincoln, Kotecky, Roman
We study equilibrium droplets in two-phase systems at parameter values corresponding to phase coexistence. Specifically, we give a self-contained microscopic derivation of the Gibbs-Thomson formula...
Rigorous analysis of discontinuous phase transitions via mean-field bounds (2003)
Abstract: We consider a variety of nearest-neighbor spin models defined on the d-dimensional hypercubic lattice Z d. Our essential assumption is that these models satisfy the condition of reflection...
Critical region for droplet formation in the two-dimensional Ising model (2002)
Biskup, Marek, Chayes, Lincoln, Kotecky, Roman
We study the formation/dissolution of equilibrium droplets in finite systems at parameters corresponding to phase coexistence. Specifically, we consider the 2D Ising model in volumes of size $L^2$,...
Rigorous analysis of discontinuous phase transitions via mean-field bounds (2002)
Biskup, Marek, Chayes, Lincoln
We consider a variety of nearest-neighbor spin models defined on the d-dimensional hypercubic lattice Z^d. Our essential assumption is that these models satisfy the condition of reflection...
On the formation/dissolution of equilibrium droplets (2002)
Biskup, Marek, Chayes, Lincoln, Kotecky, Roman
We consider liquid-vapor systems in finite volume $V\subset\R^d$ at parameter values corresponding to phase coexistence and study droplet formation due to a fixed excess $\delta N$ of particles above...
Phase transition and critical behavior in a model of organized criticality (2002)
Biskup, Marek, Blanchard, Philippe, Chayes, Lincoln, Gandolfo, Daniel, Krueger, Tyll
We study a model of ``organized'' criticality, where a single avalanche propagates through an \textit{a priori} static (i.e., organized) sandpile configuration. The latter is chosen according to an...
Abstract: We consider a variety of nearest-neighbor spin models defined on the d-dimensional hypercubic lattice Z d. Our essential assumption is that these models satisfy the condition of reflection...
Screening effect due to heavy lower tails in one-dimensional parabolic Anderson model (2000)
Biskup, Marek, Koenig, Wolfgang
We consider the large-time behavior of the solution $u\colon [0,\infty)\times\Z\to[0,\infty)$ to the parabolic Anderson problem $\partial_t u=\kappa\Delta u+\xi u$ with initial data $u(0,\cdot)=1$...
Long-time tails in the parabolic Anderson model with bounded potential (2000)
Biskup, Marek, Koenig, Wolfgang
We consider the parabolic Anderson problem $\partial_t u=\kappa\Delta u+\xi u$ on $(0,\infty)\times \Z^d$ with random i.i.d. potential $\xi=(\xi(z))_{z\in\Z^d}$ and the initial condition...
General Theory of Lee-Yang Zeros in Models with First-Order Phase Transitions (2000)
Biskup, Marek, Borgs, Christian, Chayes, Jennifer T., Kleinwaks, Logan J., Kotecky, Roman
We present a general, rigorous theory of Lee-Yang zeros for models with first-order phase transitions that admit convergent contour expansions. We derive formulas for the positions and the density of...
A heteropolymer near a linear interface (1999)
Biskup, Marek, Den Hollander, Frank
We consider a quenched-disordered heteropolymer, consisting of hydrophobic and hydrophylic monomers, in the vicinity of an oil-water interface. The heteropolymer is modeled by a directed simple...