Path Counting on Tree-like Graphs with a Single Entropic Trap: Critical Behavior and Finite Size Effects.

It is known that maximal entropy random walks and partition functions that count long paths on graphs tend to become localized near nodes with a high degree. Here, we revisit the simplest toy model of such a localization: a regular tree of degree p with one special node ("root") that has a degree different from all the others. We present an in-depth study of the path-counting problem precisely at the localization transition. We study paths that start from the root in both infinite trees and finite, locally tree-like regular random graphs (RRGs). For the infinite tree, we prove that the probability distribution function of the endpoints of the path is a step function. The position of the step moves away from the root at a constant velocity v=(p-2)/p. We find the width and asymptotic shape of the distribution in the vicinity of the shock. For a finite RRG, we show that a critical slowdown takes place, and the trajectory length needed to reach the equilibrium distribution is on the order of N instead of logp-1N away from the transition. We calculate the exact values of the equilibrium distribution and relaxation length, as well as the shapes of slowly relaxing modes.

A statistical model of intra-chromosome contact maps.

A statistical model describing a fine structure of the intra-chromosome maps obtained by a genome-wide chromosome conformation capture method (Hi-C) is proposed. The model combines hierarchical chain folding with a quenched heteropolymer structure of primary chromatin sequences. It is conjectured that the observed Hi-C maps are statistical averages over many different ways of hierarchical genome folding. It is shown that the existence of a quenched primary structure coupled with hierarchical folding induces a full range of features observed in experimental Hi-C maps: hierarchical elements, chess-board intermittency and large-scale compartmentalization.

How a viscoelastic solution of wormlike micelles transforms into a microemulsion upon absorption of hydrocarbon: new insight.

In this article, we investigate the effect of hydrocarbon addition on the rheological properties and structure of wormlike micellar solutions of potassium oleate. We show that a viscoelastic solution of entangled micellar chains is extremely responsive to hydrocarbons-the addition of only 0.5 wt % n-dodecane results in a drastic drop in viscosity by up to 5 orders of magnitude, which is due to the complete disruption of micelles and the formation of microemulsion droplets. We study the whole range of the transition of wormlike micelles into microemulsion droplets and discover that it can be divided into three regions: (i) in the first region, the solutions retain a high viscosity (∼10-350 Pa·s), the micelles are entangled but their length is reduced by the solubilization of hydrocarbons; (ii) in the second region, the system transitions to the unentangled regime and the viscosity sharply decreases as a result of further micelle shortening and the appearance of microemulsion droplets; (iii) in the third region, the viscosity is low (∼0.001 Pa·s) and only microemulsion droplets remain in the solution. The experimental studies were accompanied by theoretical considerations, which allowed us to reveal for the first time that (i) one of the leading mechanisms of micelle shortening is the preferential accumulation of the solubilized hydrocarbon in the spherical end caps of wormlike micelles, which makes the end caps thermodynamically more favorable; (ii) the onset of the sharp drop in viscosity is correlated with the crossover from the entangled to unentangled regime of the wormlike micellar solution taking place upon the shortening of micellar chains; and (iii) in the unentangled regime short cylindrical micelles coexist with microemulsion droplets.

Phase transition in random planar diagrams and RNA-type matching.

We study the planar matching problem, defined by a symmetric random matrix with independent identically distributed entries, taking values zero and one. We show that the existence of a perfect planar matching structure is possible only above a certain critical density, p(c), of allowed contacts (i.e., of ones). Using a formulation of the problem in terms of Dyck paths and a matrix model of planar contact structures, we provide an analytical estimation for the value of the transition point, p(c), in the thermodynamic limit. This estimation is close to the critical value, p(c)≈0.379, obtained in numerical simulations based on an exact dynamical programming algorithm. We characterize the corresponding critical behavior of the model and discuss the relation of the perfect-imperfect matching transition to the known molten-glass transition in the context of random RNA secondary structure formation. In particular, we provide strong evidence supporting the conjecture that the molten-glass transition at T=0 occurs at p(c).

Number of common sites visited by N random walkers.

We compute analytically the mean number of common sites, W(N)(t), visited by N independent random walkers each of length t and all starting at the origin at t = 0 in d dimensions. We show that in the (N-d) plane, there are three distinct regimes for the asymptotic large-t growth of W(N)(t). These three regimes are separated by two critical lines d = 2 and d = d(c)(N) = 2N/(N-1) in the (N-d) plane. For d<2, W(N) (t) ~ t(d/2) for large t (the N dependence is only in the prefactor). For 2 < d < d(c)(N), W(N)(t) ~ t(ν) where the exponent ν = N-d(N-1)/2 varies with N and d. For d > d(c)(N), W(N)(t) → const as t → ∞. Exactly at the critical dimensions there are logarithmic corrections: for d=2, we get W(N)(t) ~ t/[ln t](N), while for d = d(c)(N), W(N)(t) ~ ln t for large t. Our analytical predictions are verified in numerical simulations.