Topological transition in disordered planar matching: combinatorial arcs expansion

In this paper, we investigate analytically the properties of the disordered Bernoulli model of planar matching. This model is characterized by a topological phase transition, yielding complete planar matching solutions only above a critical density threshold. We develop a combinatorial procedure of arcs expansion that explicitly takes into account the contribution of short arcs, and allows to obtain an accurate analytical estimation of the critical value by reducing the global constrained problem to a set of local ones. As an application to a toy representation of the RNA secondary structures, we suggest generalized models that incorporate a one-to-one correspondence between the contact matrix and the RNA-type sequence, thus giving sense to the notion of effective non-integer alphabets.Comment: 28 pages, 6 figures, published versio

Ballistic deposition patterns beneath a growing KPZ interface

We consider a (1+1)-dimensional ballistic deposition process with next-nearest neighbor interaction, which belongs to the KPZ universality class, and introduce for this discrete model a variational formulation similar to that for the randomly forced continuous Burgers equation. This allows to identify the characteristic structures in the bulk of a growing aggregate ("clusters" and "crevices") with minimizers and shocks in the Burgers turbulence, and to introduce a new kind of equipped Airy process for ballistic growth. We dub it the "hairy Airy process" and investigate its statistics numerically. We also identify scaling laws that characterize the ballistic deposition patterns in the bulk: the law of "thinning" of the forest of clusters with increasing height, the law of transversal fluctuations of cluster boundaries, and the size distribution of clusters. The corresponding critical exponents are determined exactly based on the analogy with the Burgers turbulence and simple scaling considerations.Comment: 10 pages, 5 figures. Minor edits: typo corrected, added explanation of two acronyms. The text is essentially equivalent to version

Random Operator Approach for Word Enumeration in Braid Groups

We investigate analytically the problem of enumeration of nonequivalent primitive words in the braid group B_n for n >> 1 by analysing the random word statistics and the target space on the basis of the locally free group approximation. We develop a "symbolic dynamics" method for exact word enumeration in locally free groups and bring arguments in support of the conjecture that the number of very long primitive words in the braid group is not sensitive to the precise local commutation relations. We consider the connection of these problems with the conventional random operator theory, localization phenomena and statistics of systems with quenched disorder. Also we discuss the relation of the particular problems of random operator theory to the theory of modular functionsComment: 36 pages, LaTeX, 4 separated Postscript figures, submitted to Nucl. Phys. B [PM

Adsorption of a random heteropolymer at a potential well revisited: location of transition point and design of sequences

The adsorption of an ideal heteropolymer loop at a potential point well is investigated within the frameworks of a standard random matrix theory. On the basis of semi-analytical/semi-numerical approach the histogram of transition points for the ensemble of quenched heteropolymer structures with bimodal symmetric distribution of types of chain's links is constructed. It is shown that the sequences having the transition points in the tail of the histogram display the correlations between nearest-neighbor monomers.Comment: 11 pages (revtex), 3 figure

An Anisotropic Ballistic Deposition Model with Links to the Ulam Problem and the Tracy-Widom Distribution

We compute exactly the asymptotic distribution of scaled height in a (1+1)--dimensional anisotropic ballistic deposition model by mapping it to the Ulam problem of finding the longest nondecreasing subsequence in a random sequence of integers. Using the known results for the Ulam problem, we show that the scaled height in our model has the Tracy-Widom distribution appearing in the theory of random matrices near the edges of the spectrum. Our result supports the hypothesis that various growth models in $(1+1)$ dimensions that belong to the Kardar-Parisi-Zhang universality class perhaps all share the same universal Tracy-Widom distribution for the suitably scaled height variables.Comment: 5 pages Revtex, 3 .eps figures included, new references adde

Localization in simple multiparticle catalytic absorption model

We consider the phase transition in the system of n simultaneously developing random walks on the halfline x>=0. All walks are independent on each others in all points except the origin x=0, where the point well is located. The well depth depends on the number of particles simultaneously staying at x=0. We consider the limit n>>1 and show that if the depth growth faster than 3/2 n ln(n) with n, then all random walks become localized simultaneously at the origin. In conclusion we discuss the connection of that problem with the phase transition in the copolymer chain with quenched random sequence of monomers considered in the frameworks of replica approach.Comment: 17 pages in LaTeX, 5 PostScript figures; submitted to J.Phys.(A): Math. Ge

Différents aspects de la physique statistique de systèmes avec contraintes topologiques

Ce travail présente différentes méthodes permettant une étude statistique de systèmes avec contraintes topologiques. Des exemples de telles contraintes issus de domaines variés sont exposés. Dans le cas de systèmes à géométrie linéaire comme les polymères, la modélisation proposée mène à l'étude de processus stochastiques sur le groupe fondamentale de l'espace des configu- rations. Deux modèles, décrivant différents cas d'emmêlements de tels objets linéaires sont exposés en détails. Le premier, RWAO, décrit un polymère évoluant dans un espace avec obstacles. Le second traite d'objets dirigés et se ramène à l'étude de marches aléatoires sur le groupe de tresses B3. La résolution du problème de marche aléatoire sur groupe associé à chacun de ces modèles donne accès à différentes quantités physiquement pertinentes. La géométrie hyperbolique sous-jacente à ces groupes d'homotopie est étudiée, et permet en particulier la définition et le calcul de la distribution d'un inva- riant topologique d'origine géométrique. L'étude des espaces de recouvrement est menée pour différents modèles, et permet de définir analytiquement des surfaces aux propriétés multifractales intéressantesThis work presents different methods allowing a statistical study of sys- tems with topological constraints. Exemples of such constraints coming from different fields of science are exposed. ln the case of linear abjects such as polymers, the model proposed here leads to the study of stochastic pro cesses on the fundamental group of the configuration space. Two models describing different types of entanglements of such linear abjects are exposed in details. The first one, RWAO, describes a polymer living in a space with obstacles. The second one deals with directed abjects, and is modelized by random walks on braid groups En. Resolution of the random walk on group problem associated with each of these models gives access to different quantities of physical interest. The hyperbolic geometry underlying these homotopy groups is studied, and allows in particular definition and computation of a new "geo- metrical" topological invariant. The study of covering spaces is tackeled, and leads to surfaces with interesting multifractal properties.ORSAY-PARIS 11-BU Sciences (914712101) / SudocSudocFranceF