A Droplet within the Spherical Model

Various substances in the liquid state tend to form droplets. In this paper the shape of such droplets is investigated within the spherical model of a lattice gas. We show that in this case the droplet boundary is always diffusive, as opposed to sharp, and find the corresponding density profiles (droplet shapes). Translation-invariant versions of the spherical model do not fix the spatial location of the droplet, hence lead to mixed phases. To obtain pure macroscopic states (which describe localized droplets) we use generalized quasi-averaging. Conventional quasi-averaging deforms droplets and, hence, can not be used for this purpose. On the contrary, application of the generalized method of quasi-averages yields droplet shapes which do not depend on the magnitude of the applied external field.Comment: 22 pages, 2 figure

Random walks in random environment with Markov dependence on time

We consider a simple model of discrete-time random walk on Zν, ν = 1, 2, . . . in a random environment independent in space and with Markov evolution in time. We focus on the application of methods based on the properties of the transfer matrix and on spectral analysis. In section 2 we give a new simple proof of the existence of invariant subspaces, with an explicit condition on the parameters. The remaining part is devoted to a review of the results obtained so far for the quenched random walk and the environment from the point of view of the random walk, with a brief discussion of the methods.Ми розглядаємо просту модель випадкового блукання з дискретним часом у Zν, ν = 1, 2, . . . у випадковому середовищi, що є незалежним у просторi i має маркiвську еволюцiю у часi. Ми зосереджуємось на застосуваннi методiв, що ґрунтуються на властивостях трансфер-матрицi i на спектральному аналiзi. У §2 ми подаємо просте доведення iснування iнварiантних пiдпросторiв, що використовує явну умову для параметрiв. Решта роботи присвячується огляду результатiв одержаних дотепер для замороженого випадкового блукання i оточення з точки зору випадкового блукання, а також короткому обговоренню методiв

Relaxation times for Hamiltonian systems

Usually, the relaxation times of a gas are estimated in the frame of the Boltzmann equation. In this paper, instead, we deal with the relaxation problem in the frame of the dynamical theory of Hamiltonian systems, in which the definition itself of a relaxation time is an open question. We introduce a lower bound for the relaxation time, and give a general theorem for estimating it. Then we give an application to a concrete model of an interacting gas, in which the lower bound turns out to be of the order of magnitude of the relaxation times observed in dilute gases.Comment: 26 page

Short-time Gibbsianness for Infinite-dimensional Diffusions with Space-Time Interaction

We consider a class of infinite-dimensional diffusions where the interaction between the components is both spatial and temporal. We start the system from a Gibbs measure with finite-range uniformly bounded interaction. Under suitable conditions on the drift, we prove that there exists $t_0>0$ such that the distribution at time $t\leq t_0$ is a Gibbs measure with absolutely summable interaction. The main tool is a cluster expansion of both the initial interaction and certain time-reversed Girsanov factors coming from the dynamics

Large Deviations in the Superstable Weakly Imperfect Bose Gas

The superstable Weakly Imperfect Bose Gas {(WIBG)} was originally derived to solve the inconsistency of the Bogoliubov theory of superfluidity. Its grand-canonical thermodynamics was recently solved but not at {point of} the {(first order)} phase transition. This paper proposes to close this gap by using the large deviations formalism and in particular the analysis of the Kac distribution function. It turns out that, as a function of the chemical potential, the discontinuity of the Bose condensate density at the phase transition {point} disappears as a function of the particle density. Indeed, the Bose condensate continuously starts at the first critical particle density and progressively grows but the free-energy per particle stays constant until the second critical density is reached. At higher particle densities, the Bose condensate density as well as the free-energy per particle both increase {monotonously}

A Contour Method on Cayley tree

We consider a finite range lattice models on Cayley tree with two basic properties: the existence of only a finite number of ground states and with Peierls type condition. We define notion of a contour for the model on the Cayley tree. By a contour argument we show the existence of $s$ different (where $s$ is the number of ground states) Gibbs measures.Comment: 12 page

The ^4He trimer as an Efimov system

We review the results obtained in the last four decades which demonstrate the Efimov nature of the $^4$He three-atomic system.Comment: Review article for a special issue of the Few-Body Systems journal devoted to Efimov physic

Directed polymers in Markov random media

We consider a model of directed polymers in discrete space and time assuming a Markov dependence of the environment in time. We extend results on the almost-sure validity of the central limit theorem for small randomness in space dimension 3 which were previously obtained for an independent environment by relying on two main technical tools: the analysis of the spectrum of a kind of transfer matrix which allows one to treat the averaged model, and the explicit construction of a multiplicative orthonormal basis in the appropriate L_2 space, together with cluster estimates of cumulants of the basis functions

Random walks in quenched i.i.d. space-time random environment are always a.s. diffusive

We consider a general model of discrete-time random walk X-t on the lattice (nu), nu = 1,..., in a random environment xi={xi(t,x):(t,x)is an element of(nu+1)} with i.i.d. components xi(t,x). Previous results on the a.s. validity of the Central Limit Theorem for the quenched model required a small stochasticity condition. In this paper we show that the result holds provided only that an obvious non-degeneracy condition is met. The proof is based on the analysis of a suitable generating function, which allows to estimate L-2 norms by contour integrals

Interacting random walk of two particles in a dynamical random environment. Decay of correlations.

This paper continues the study of a family of models studied earlier by the authors. Two particles perform discrete-time symmetric random walks on the d-dimensional integer lattice Z^d and interact locally with each other and with a random field (the “environment”) which is indexed by the lattice points. The environment evolves randomly in time its law of evolution is locally affected by the particles . The whole system is Markovian, and all interactions are assumed to be sufficiently small. It is shown that the correlations of the field at two fixed points decay in time as C t^{(−d/2)−1}. Under additional assumptions the constant C may be expressed to first order as the sum of the corresponding constants for the one-particle model. The proofs are based on the analysis of the spectrum of the system’s transition operator. The results may be extended to models containing a finite number of particles