Volume 10 (2010), Number 4. Abstracts
Yu. Bakhtin and K. Khanin. Localization and Perron–Frobenius Theory for Directed Polymers [PDF]
We consider directed polymers in a random potential given by a deterministic profile with a strong maximum at the origin taken with random sign at each integer time.
We study two main objects based on paths in this random potential. First, we use the random potential and averaging over paths to define a parabolic model via a random Feynman–Kac evolution operator. We show that for the resulting cocycle, there is a unique positive cocycle eigenfunction serving as a forward and pullback attractor. Secondly, we use the potential to define a Gibbs specification on paths for any bounded time interval in the usual way and study the thermodynamic limit and existence and uniqueness of an infinite volume Gibbs measure. Both main results claim that the local structure of interaction leads to a unique macroscopic object for almost every realization of the random potential.
Keywords. Directed polymers, localization, Perron–Frobenius theorem, parabolic model.
2000 Mathematics Subject Classification. 82D30, 82D60.
A.C.D. van Enter, R. Fernández, F. den Hollander, and F. Redig. A Large-Deviation View on Dynamical Gibbs-Non-Gibbs Transitions [PDF]
We develop a space-time large-deviation point of view on Gibbs-non-Gibbs transitions in spin systems subject to a stochastic spin-flip dynamics. Using the general theory for large deviations of functionals of Markov processes outlined in a recent book by Feng and Kurtz, we show that the trajectory under the spin-flip dynamics of the empirical measure of the spins in a large block in Zd satisfies a large deviation principle in the limit as the block size tends to infinity. The associated rate function can be computed as the action functional of a Lagrangian that is the Legendre transform of a certain non-linear generator, playing a role analogous to the moment-generating function in the Gärtner–Ellis theorem of large deviation theory when this is applied to finite-dimensional Markov processes. This rate function is used to define the notion of “bad empirical measures”, which are the discontinuity points of the optimal trajectories (i.e., the trajectories minimizing the rate function) given the empirical measure at the end of the trajectory. The dynamical Gibbs-non-Gibbs transitions are linked to the occurrence of bad empirical measures: for short times no bad empirical measures occur, while for intermediate and large times bad empirical measures are possible. A future research program is proposed to classify the various possible scenarios behind this crossover, which we refer to as a “nature-versus-nurture” transition.
Keywords. Stochastic spin-flip dynamics, Gibbs-non-Gibbs transition, empirical measure, non-linear generator, Nisio control semigroup, large deviation principle, bad configurations, bad empirical measures, nature versus nurture.
2000 Mathematics Subject Classification. Primary: 60F10, 60G60, 60K35; Secondary: 82B26, 82C22.
W. Faris. Combinatorial Species and Cluster Expansions [PDF]
This paper will survey recent progress on clarifying the connection between enumerative combinatorics and cluster expansions. The combinatorics side concerns species of combinatorial structures and the associated exponential generating functions. Cluster expansions, on the other hand, are supposed to give convergent expressions for measures on infinite dimensional spaces, such as those that occur in statistical mechanics. There is a dictionary between these two subjects that sheds light on each of them. In particular, it gives insight into convergence results for cluster expansions, including a well-known result of Roland Dobrushin. Furthermore, the species framework provides a context for recent results of Fernández–Procacci and of the author.
Keywords. Combinatorial species, species of structures, exponential generating function, equilibrium statistical mechanics, grand partition function, cluster expansion.
2000 Mathematics Subject Classification. Primary: 60K35, 82B05, 05A15; Secondary: 82B20, 05C30.
J. Fritz. Application of Relaxation Schemes in the Microscopic Theory of Hydrodynamics [PDF]
We consider stochastic evolution of particles moving on Z with opposite speeds. This model of interacting exclusions admits a hyperbolic (Euler) scaling, and its hydrodynamic limit results in the Leroux system of PDE theory. The basic model can be modified by introducing a spin-flip, or a creation-annihilation mechanism. In a regime of shock waves the method of compensated compactness is applied. We are going to discuss usefulness of another tool of the theory of conservation laws, the technique of relaxation schemes is extended to microscopic systems.
Keywords. Interacting exclusions, hyperbolic scaling, Lax entropy pairs, compensated compactness, logarithmic Sobolev inequalities, relaxation schemes.
2000 Mathematics Subject Classification. Primary: 60K31; Secondary: 82C22.
P. Major. Estimation of Multiple Random Integrals and U-Statistics [PDF]
Here I give a short survey of some problems on multiple random integrals and U-statistics together with some other questions which occur during their study in a natural way. I write down the main results, discuss their background together with some pictures and mathematical ideas which explain them better.
Keywords. Wiener–Itô integrals, degenerate U-statistics, large deviation results.
2000 Mathematics Subject Classification. 60E15.
O. Ogievetsky and V. Schechtman. Nombres de Bernoulli et une formule de Schlömilch–Ramanujan [PDF]
Nous discutons quelques formules qui utilisent les nombres de Bernoulli. Dans la première partie de cet article, on établit un lien étroit entre la formule d'Euler–Maclaurin et l'équation fonctionelle de Rota–Baxter. Dans la deuxième partie, on présente une simple démonstration d'une formule de Schlömilch–Ramanujan sur la sommation de certaines séries exponentielles, formant une famille à un paramètre naturel impair l. Un phénomène surprenant est observé : pour ces séries, l'approximation d'Euler–Maclaurin (de la somme par l'intégrale) est exacte si l>1.
Keywords. Bernoulli numbers, Euler–Maclaurin formula, Rota–Baxter equation, Dedekind function, Schlömilch formula, Ramanujan formula, Eisenstein series, Weierstrass function.
2000 Mathematics Subject Classification. 11B68, 65B15, 11F03.
E. Pechersky, E. Petrova, and S. Pirogov. Phase Transitions of Laminated Models at any Temperature [PDF]
The standard Pirogov–Sinai theory is generalized to the class of models with two modes of interaction: longitudinal and transversal. Under rather general assumptions about the longitudinal interaction and for one specific form of the transversal interaction it is proved that such system has a variety of phase transitions at any temperature: the parameter which plays the role of inverse temperature is the strength of the transversal interaction. The concrete examples of such systems are (1+1)-dimensional models.
Keywords. Phase transitions, lattice models, Pirogov–Sinai theory.
2000 Mathematics Subject Classification. 82B20, 82B26.
L. Shepp. Probability Problems of Dobrushin Type [PDF]
I review five probability problem areas which I believe are in the spirit of my friend, Roland. Were he still around, he would shed light on them in his inimitable, insightful way. The novelty of this note is that it ties together a lot of my special papers in a way that perhaps only Roland would grasp; even though I cannot see the commonality. Anyone who wants to comment is more than welcome to do so: email@example.com
Keywords. Probability, math.
2000 Mathematics Subject Classification. 60G60.
E. Verbitskiy. Variational Principle for Fuzzy Gibbs Measures [PDF]
In this paper we study a large class of renormalization transformations of measures on lattices. An image of a Gibbs measure under such transformation is called a fuzzy Gibbs measure. Transformations of this type and fuzzy Gibbs measures appear naturally in many fields. Examples include the hidden Markov processes (HMP), memoryless channels in information theory, continuous block factors of symbolic dynamical systems, and many renormalization transformations of statistical mechanics. The main result is the generalization of the classical variational principle of Dobrushin–Lanford–Ruelle for Gibbs measures to the class of fuzzy Gibbs measures.
Keywords. Non-Gibbsian measures, renormalization, deterministic and random transformations, variational principle.
2000 Mathematics Subject Classification. Primary: 82B20; Secondary: 82B28, 37B10, 37A60.
Moscow Mathematical Journal