arXiv subjects

Marek Biskup

Publications and source records attributed to Marek Biskup.

At least 55 records · Page 3Linked to original sources

arXiv2012

Recent progress on the Random Conductance Model

Recent progress on the understanding of the Random Conductance Model is reviewed and commented. A particular emphasis is on the results on the scaling limit of the random walk among random conductances for almost every realization of the environment, observations on the behavior of the effective resistance as well as the scaling limit of certain models of gradient fields with non-convex interactions. The text is an expanded version of the lecture notes for a course delivered at the 2011 Cornell Summer School on Probability.

math.PR↗

arXiv2011

Absence of magnetism in continuous-spin systems with long-range antialigning forces

We consider continuous-spin models on the $d$-dimensional hypercubic lattice with the spins $σ_x$ \emph{a priori} uniformly distributed over the unit sphere in $\R^n$ (with $n\ge2$) and the interaction energy having two parts: a short-range part, represented by a potential $Φ$, and a long-range antiferromagnetic part $λ|x-y|^{-s}σ_x\cdotσ_y$ for some exponent $s>d$ and $λ\ge0$. We assume that $Φ$ is twice continuously differentiable, finite range and invariant under rigid rotations of all spins. For $d\ge1$, $s\in(d,d+2]$ and any $λ>0$, we then show that the expectation of each $σ_x$ vanishes in all translation-invariant Gibbs states. In particular, the spontaneous magnetization is zero and block-spin averages vanish in all (translation invariant or not) Gibbs states. This contrasts the situation of $λ=0$ where the ferromagnetic nearest-neighbor systems in $d\ge3$ exhibit strong magnetic order at sufficiently low temperatures. Our theorem extends an earlier result of A. van Enter ruling out magnetized states with uniformly positive two-point correlation functions.

math-ph↗

arXiv2010

Scaling limit for a class of gradient fields with nonconvex potentials

We consider gradient fields $(ϕ_x:x\in \mathbb{Z}^d)$ whose law takes the Gibbs--Boltzmann form $Z^{-1}\exp\{-\sum_{< x,y>}V(ϕ_y-ϕ_x)\}$, where the sum runs over nearest neighbors. We assume that the potential $V$ admits the representation \[V(η):=-\log\int\varrho({d}κ)\exp\biggl[-{1/2}κ\et a^2\biggr],\] where $\varrho$ is a positive measure with compact support in $(0,\infty)$. Hence, the potential $V$ is symmetric, but nonconvex in general. While for strictly convex $V$'s, the translation-invariant, ergodic gradient Gibbs measures are completely characterized by their tilt, a nonconvex potential as above may lead to several ergodic gradient Gibbs measures with zero tilt. Still, every ergodic, zero-tilt gradient Gibbs measure for the potential $V$ above scales to a Gaussian free field.

math.PR↗

arXiv2010

True nature of long-range order in a plaquette orbital model

We analyze the classical version of a plaquette orbital model that was recently introduced and studied numerically by S. Wenzel and W. Janke. In this model, edges of the square lattice are partitioned into $x$ and $z$-types that alternate along both coordinate directions and thus arrange into a checkerboard pattern of $x$ and $z$-plaquettes; classical O(2)-spins are then coupled ferromagnetically via their first components over the $x$-edges and via their second components over the $z$-edges. We prove from first principles that, at sufficiently low temperatures, the model exhibits orientational long-range order (OLRO) in one of the two principal lattice directions. Magnetic order is precluded by the underlying symmetries. A similar set of results is inferred also for quantum systems with large spin although the $\frac12$ instance currently seems beyond the reach of rigorous methods. We point out that the Neél order in the plaquette energy distribution observed in numerical simulations is an artefact of the OLRO and a judicious choice of the plaquette energies. In particular, this order seems to disappear when the plaquette energies are adjusted to vanish at the ground-state level. We also discuss the specific role of the underlying symmetries in Wenzel and Janke's simulations and propose an enhanced method of numerical sampling that could in principle significantly increase the speed of convergence.

math-ph↗

arXiv2009

Graph diameter in long-range percolation

We study the asymptotic growth of the diameter of a 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 probability decaying with the Euclidean distance as $|x-y|^{-s+o(1)}$ when $|x-y|\to\infty$. For $s\in(d,2d)$ we show that the graph diameter for the graph reduced to a box of side $L$ scales like $(\log L)^{Δ+o(1)}$ where $Δ^{-1}:=\log_2(2d/s)$. In particular, the diameter grows about as fast as the typical graph distance between two vertices at distance $L$. We also show that a ball of radius $r$ in the intrinsic metric on the (infinite) graph will roughly coincide with a ball of radius $\exp\{r^{1/Δ+o(1)}\}$ in the Euclidean metric.

math.PR↗

arXiv2009

Metastable behavior for bootstrap percolation on regular trees

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 least theta occupied neighbors, occupied sites remain occupied forever. It is known that, when b>theta>1, the limiting density q=q(p) of occupied sites exhibits a jump at some p_t=p_t(b,theta) in (0,1) from q_t:=q(p_t)<1 to q(p)=1 when p>p_t. We investigate the metastable behavior associated with this transition. Explicitly, we pick p=p_t+h with h>0 and show that, as h decreases to 0, the system lingers around the "critical" state for time order h^{-1/2} and then passes to fully occupied state in time O(1). The law of the entire configuration observed when the occupation density is q in (q_t,1) converges, as h tends to 0, to a well-defined measure.

math.PR↗

arXiv2009

Reflection positivity and phase transitions in lattice spin models

Reflection positivity (RP) is a property of Gibbs measures exhibited by a class of lattice spin systems that include the Ising, Potts and Heisenberg models. The RP property is useful because of its two basic consequences: infrared bound and chessboard estimates. These are one of basic (and rather efficient) tools for proving phase transitions in many models of physical interest. The notes presented hereby summarize the lectures on reflection positivity and its consequences that the author delivered at the Prague Summer School on Mathematical Statistical Mechanics in September 2006. The text features both the classical material on the subject from the late 1970s as well as some of the more recent developments.

math-ph↗

arXiv2007

Functional CLT for random walk among bounded random conductances

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 $ω_{xy}\in[0,1]$. Apart from the requirement that the bonds with positive conductances percolate, we pose no restriction on the law of the $ω$'s. We prove that, for a.e. realization of the environment, the path distribution of the walk converges weakly to that of non-degenerate, isotropic Brownian motion. The quenched functional CLT holds despite the fact that the local CLT may fail in $d\ge5$ due to anomalously slow decay of the probability that the walk returns to the starting point at a given time (cf math.PR/0611666).

math.PR↗

arXiv2007

Anomalous heat-kernel decay for random walk among bounded random conductances

We consider the nearest-neighbor simple random walk on $\Z^d$, $d\ge2$, driven by a field of bounded random conductances $ω_{xy}\in[0,1]$. The conductance law is i.i.d. subject to the condition that the probability of $ω_{xy}>0$ exceeds the threshold for bond percolation on $\Z^d$. For environments in which the origin is connected to infinity by bonds with positive conductances, we study the decay of the $2n$-step return probability $P_ω^{2n}(0,0)$. We prove that $P_ω^{2n}(0,0)$ is bounded by a random constant times $n^{-d/2}$ in $d=2,3$, while it is $o(n^{-2})$ in $d\ge5$ and $O(n^{-2}\log n)$ in $d=4$. By producing examples with anomalous heat-kernel decay approaching $1/n^2$ we prove that the $o(n^{-2})$ bound in $d\ge5$ is the best possible. We also construct natural $n$-dependent environments that exhibit the extra $\log n$ factor in $d=4$. See also math.PR/0701248.

math.PR↗

arXiv2006

On the absence of ferromagnetism in typical 2D ferromagnets

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 distance. The physical context of such models is discussed; primarily this is $d=2$ and $s=3$ where, at long distances, genuine magnetic interactions between genuine magnetic dipoles are of this form. We prove that when the power of decay lies above $d$ and does not exceed $d+1$, then for all temperatures, the spontaneous magnetization is zero. In contrast, we also show that for powers exceeding $d+1$ (with $d\ge2$) magnetic order can occur.

math-ph↗

arXiv2006

Quantum spin systems at positive temperature

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 prove a phase transition in the classical model, the corresponding quantum model will have a similar phase transition, provided the inverse temperature $β$ and the magnitude of the quantum spins $\CalS$ satisfy $β\ll\sqrt\CalS$. From the quantum system we require that it is reflection positive and that it has a meaningful classical limit; the core technical estimate may be described as an extension of the Berezin-Lieb inequalities down to the level of matrix elements. The general theory is applied to prove phase transitions in various quantum spin systems with $\CalS\gg1$. The most notable examples are the quantum orbital-compass model on $\Z^2$ and the quantum 120-degree model on $\Z^3$ which are shown to exhibit symmetry breaking at low-temperatures despite the infinite degeneracy of their (classical) ground state.

math-ph↗

arXiv2006

Large-deviations/thermodynamic approach to percolation on the complete graph

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 the giant component occupies a fixed fraction of the graph while all other components are ``small.'' One consequence is an immediate derivation of the ``cavity'' formula for the fraction of vertices in the giant component. As a by-product of our analysis we compute the large-deviation rate functions for the probability of the event that the random graph is connected, the event that it contains no cycles and the event that it contains only ``small'' components.

math.PR↗

arXiv2006

Mean-field driven first-order phase transitions in systems with long-range interactions

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 sense that the $J_{x,y}$'s exhibit either exponential or power-law fall-off. Under the technical condition of reflection positivity and for sufficiently spread out interactions, we prove that the model exhibits a first-order phase transition whenever the associated mean-field theory signals such a transition. As a consequence, e.g., in dimensions $d\ge3$, we can finally provide examples of the 3-state Potts model with spread-out, exponentially decaying interactions, which undergoes a first-order phase transition as the temperature varies. Similar transitions are established in dimensions $d=1,2$ for power-law decaying interactions and in high dimensions for next-nearest neighbor couplings. In addition, we also investigate the limit of infinitely spread-out interactions. Specifically, we show that once the mean-field theory is in a unique ``state,'' then in any sequence of translation-invariant Gibbs states various observables converge to their mean-field values and the states themselves converge to a product measure.

math-ph↗

arXiv2006

Phase coexistence of gradient Gibbs states

We consider the (scalar) gradient fields $η=(η_b)$--with $b$ denoting the nearest-neighbor edges in $\Z^2$--that are distributed according to the Gibbs measure proportional to $\texte^{-βH(η)}ν(\textdη)$. Here $H=\sum_bV(η_b)$ is the Hamiltonian, $V$ is a symmetric potential, $β>0$ is the inverse temperature, and $ν$ is the Lebesgue measure on the linear space defined by imposing the loop condition $η_{b_1}+η_{b_2}=η_{b_3}+η_{b_4}$ for each plaquette $(b_1,b_2,b_3,b_4)$ in $\Z^2$. For convex $V$, Funaki and Spohn have shown that ergodic infinite-volume Gibbs measures are characterized by their tilt. We describe a mechanism by which the gradient Gibbs measures with non-convex $V$ undergo a structural, order-disorder phase transition at some intermediate value of inverse temperature $β$. At the transition point, there are at least two distinct gradient measures with zero tilt, i.e., $E η_b=0$.

math.PR↗

arXiv2006

Quenched invariance principle for simple random walk on percolation clusters

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 distribution of the walk converges weakly to that of non-degenerate, isotropic Brownian motion. Our analysis is based on the consideration of a harmonic deformation of the infinite cluster on which the random walk becomes a square-integrable martingale. The size of the deformation, expressed by the so called corrector, is estimated by means of ergodicity arguments.

math.PR↗

arXiv2005

Forbidden gap argument for phase transitions proved by means of chessboard estimates

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 diagram where more than one Gibbs states coexist, but that it can also be used to rule out the existence of shift-ergodic states that differ significantly from those proved to exist. For models depending on a parameter (say, the temperature), this shows that the values of the conjugate thermodynamic quantity (the energy) inside the "transitional gap" are forbidden in all shift-ergodic Gibbs states. We point out several models where our result provides useful additional information concerning the set of possible thermodynamic equilibria.

math-ph↗

arXiv2005

On the scaling of the chemical distance in long-range percolation models

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 |x-y|\to\infty, where s>0 and |\cdot| is a norm distance on Z^d, and supposing that the resulting random graph contains an infinite connected component C_{\infty}, we let D(x,y) be the graph distance between x and y measured on C_{\infty}. Our main result is that, for s\in(d,2d), D(x,y)=(\log|x-y|)^{Δ+o(1)},\qquad x,y\in C_{\infty}, |x-y|\to\infty, where Δ^{-1} is the binary logarithm of 2d/s and o(1) is a quantity tending to zero in probability as |x-y|\to\infty. Besides its interest for general percolation theory, this result sheds some light on a question that has recently surfaced in the context of ``small-world'' phenomena. As part of the proof we also establish tight bounds on the probability that the largest connected component in a finite box contains a positive fraction of all sites in the box.

math.PR↗

arXiv2005

Colligative properties of solutions: I. Fixed concentrations

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 model of a solvent-solute system and show that, in the ensemble with a fixed amount of solute, a macroscopic phase separation occurs in an interval of values of the chemical potential of the solvent. The boundaries of the phase separation domain in the phase diagram are characterized and shown to asymptotically agree with the formulas used in heuristic analyses of freezing point depression. The limit of infinitesimal concentrations is described in a subsequent paper.

math-ph↗