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Marek Biskup

Publications and source records attributed to Marek Biskup.

69 records · Page 4Linked to original sources
arXiv2005

Colligative properties of solutions: II. Vanishing concentrations

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 potential scale like $c=ξ/L$ and $h=b/L$, respectively. We find that there exists a critical value $\xit$ such that no phase separation occurs for $ξ\le\xit$ while, for $ξ>\xit$, the two phases of the solvent coexist for an interval of values of $b$. Moreover, phase separation begins abruptly in the sense that a macroscopic fraction of the system suddenly freezes (or melts) forming a crystal (or droplet) of the complementary phase when $b$ reaches a critical value. For certain values of system parameters, under ``frozen'' boundary conditions, phase separation also ends abruptly in the sense that the equilibrium droplet grows continuously with increasing $b$ and then suddenly jumps in size to subsume the entire system. Our findings indicate that the onset of freezing-point depression is in fact a surface phenomenon.

math-ph↗
arXiv2004

Orbital ordering in transition-metal compounds: I. The 120-degree model

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 of the spins gets coupled in each coordinate direction. Although the Hamiltonian has only discrete symmetries, it turns out that every constant field is a ground state. Employing a combination of spin-wave and contour arguments we establish the existence of long-range order at low temperatures. This suggests a mechanism for a type of ordering in certain models of transition-metal compounds where the very existence of long-range order has heretofore been a matter of some controversy.

cond-mat.stat-mech↗
arXiv2004

Orbital order in classical models of transition-metal compounds

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 demonstrate that at low temperatures these models exhibit a long-range order which arises via an "order by disorder" mechanism. This strongly indicates that there is orbital ordering in the quantum version of these models, notwithstanding recent rigorous results on the absence of spin order in these systems.

cond-mat.stat-mech↗
arXiv2004

Order by disorder, without order, in a two-dimensional spin system with O(2) symmetry

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 states distinguished by local order among columns or, respectively, rows. Overall, there is no magnetic order in accord with the classic Mermin-Wagner theorem. The method of proof employs a rigorous version of "order by disorder," whereby a high degeneracy among the ground states is lifted according to the differences in their associated spin-wave spectra.

math-ph↗
arXiv2004

Partition function zeros at first-order phase transitions: A general analysis

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 large class of spin models in a companion paper [BBCKK2, math-ph/0304007]. Under these assumptions, we derive equations whose solutions give the location of the zeros of the partition function with periodic boundary conditions, up to an error which we prove is (generically) exponentially small in the linear size of the system. For asymptotically large systems, the zeros concentrate on phase boundaries which are simple curves ending in multiple points. For models with an Ising-like plus-minus symmetry, we also establish a local version of the Lee-Yang Circle Theorem. This result allows us to control situations when in one region of the complex plane the zeros lie precisely on the unit circle, while in the complement of this region the zeros concentrate on less symmetric curves.

math-ph↗
arXiv2004

Partition function zeros at first-order phase transitions: Pirogov-Sinai theory

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 assumptions under which the results of [BBCKK] were established are satisfied by a large class of lattice models. These models are characterized by two basic properties: The existence of only a finite number of ground states and the availability of an appropriate contour representation. This setting includes, for instance, the Ising, Potts and Blume-Capel models at low temperatures. The combined results of [BBCKK] and the present paper provide complete control of the zeros of the partition function with periodic boundary conditions for all models in the above class.

math-ph↗
arXiv2003

Long-time tails in the parabolic Anderson model with bounded potential

We consider the parabolic Anderson problem $\partial_t u=κΔu+ξu$ on $(0,\infty)\times \Z^d$ with random i.i.d. potential $ξ=(ξ(z))_{z\in\Z^d}$ and the initial condition $u(0,\cdot)\equiv1$. Our main assumption is that $\esssupξ(0)=0$. Depending on the thickness of the distribution $\prob(ξ(0)\in\cdot)$ close to its essential supremum, we identify both the asymptotics of the moments of $u(t,0)$ and the almost-sure asymptotics of $u(t,0)$ as $t\to\infty$ in terms of variational problems. As a by-product, we establish Lifshitz tails for the random Schrödinger operator $-κΔ-ξ$ at the bottom of its spectrum. In our class of $ξ$ distributions, the Lifshitz exponent ranges from $d/2$ to $\infty$; the power law is typically accompanied by lower-order corrections.

math-ph↗
arXiv2003

Rigorous analysis of discontinuous phase transitions via mean-field bounds

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 positivity. We prove that whenever the associated mean-field theory predicts a discontinuous transition, the actual model also undergoes a discontinuous transition (which occurs near the mean-field transition temperature), provided the dimension is sufficiently large or the first-order transition in the mean-field model is sufficiently strong. As an application of our general theory, we show that for d sufficiently large, the 3-state Potts ferromagnet on Z^d undergoes a first-order phase transition as the temperature varies. Similar results are established for all q-state Potts models with q>=3, the r-component cubic models with r>=4 and the O(N)-nematic liquid-crystal models with N>=3.

math.PR↗
arXiv2003

Critical region for droplet formation in the two-dimensional Ising model

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 $L^2$, inverse temperature $β>\betac$ and overall magnetization conditioned to take the value $\mstar L^2-2\mstar v_L$, where $\betac^{-1}$ is the critical temperature, $\mstar=\mstar(β)$ is the spontaneous magnetization and $v_L$ is a sequence of positive numbers. We find that the critical scaling for droplet formation/dissolution is when $v_L^{3/2} L^{-2}$ tends to a definite limit. Specifically, we identify a dimensionless parameter $Δ$, proportional to this limit, a non-trivial critical value $\Deltac$ and a function $λ_Δ$ such that the following holds: For $Δ<\Deltac$, there are no droplets beyond $\log L$ scale, while for $Δ>\Deltac$, there is a single, Wulff-shaped droplet containing a fraction $λ_Δ\ge\lamc=2/3$ of the magnetization deficit and there are no other droplets beyond the scale of $\log L$. Moreover, $λ_Δ$ and $Δ$ are related via a universal equation that apparently is independent of the details of the system.

math.PR↗
arXiv2003

Phase transition and critical behavior in a model of organized criticality

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 i.i.d. distribution from a Borel probability measure $ρ$ on $[0,1]$. The avalanche dynamics is driven by a standard toppling rule, however, we simplify the geometry by placing the problem on a directed, rooted tree. As our main result, we characterize which $ρ$ are critical in the sense that they do not admit an infinite avalanche but exhibit a power-law decay of avalanche sizes. Our analysis reveals close connections to directed site-percolation, both in the characterization of criticality and in the values of the critical exponents.

math.PR↗
arXiv2003

A proof of the Gibbs-Thomson formula in the droplet formation regime

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 for the deviation of the pressure and the density away from their equilibrium values which, according to the interpretation of the classical thermodynamics, appears due to the presence of a curved interface. The general--albeit heuristic--reasoning is corroborated by a rigorous proof in the case of the two-dimensional Ising lattice gas.

math-ph↗
arXiv2003

General Theory of Lee-Yang Zeros in Models with First-Order Phase Transitions

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 the zeros. In particular, we show that for models without symmetry, the curves on which the zeros lie are generically not circles, and can have topologically nontrivial features, such as bifurcation. Our results are illustrated in three models in a complex field: the low-temperature Ising and Blume-Capel models, and the $q$-state Potts model for $q$ large enough.

math-ph↗
arXiv2002

On the formation/dissolution of equilibrium droplets

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 $δN$ of particles above the ambient gas density. We identify a dimensionless parameter $Δ\sim(δN)^{(d+1)/d}/V$ and a \textrm{universal} value $\Deltac=\Deltac(d)$, and show that a droplet of the dense phase occurs whenever $Δ>\Deltac$, while, for $Δ<\Deltac$, the excess is entirely absorbed into the gaseous background. When the droplet first forms, it comprises a non-trivial, \textrm{universal} fraction of excess particles. Similar reasoning applies to generic two-phase systems at phase coexistence including solid/gas--where the ``droplet'' is crystalline--and polymorphic systems. A sketch of a rigorous proof for the 2D Ising lattice gas is presented; generalizations are discussed heuristically.

math-ph↗
arXiv2000

Screening effect due to heavy lower tails in one-dimensional parabolic Anderson model

We consider the large-time behavior of the solution $u\colon [0,\infty)\times\Z\to[0,\infty)$ to the parabolic Anderson problem $\partial_t u=κΔu+ξu$ with initial data $u(0,\cdot)=1$ and non-positive finite i.i.d. potentials $(ξ(z))_{z\in\Z}$. Unlike in dimensions $d\ge2$, the almost-sure decay rate of $u(t,0)$ as $t\to\infty$ is not determined solely by the upper tails of $ξ(0)$; too heavy lower tails of $ξ(0)$ accelerate the decay. The interpretation is that sites $x$ with large negative $ξ(x)$ hamper the mass flow and hence screen off the influence of more favorable regions of the potential. The phenomenon is unique to $d=1$. The result answers an open question from our previous study \cite{BK00} of this model in general dimension.

math-ph↗