17 research outputs found
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 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