The combinatorics of the leading root of the partial theta function

Recently Alan Sokal studied the leading root $x_0(q)$ of the partial theta function $\Theta_0(x,q)=\sum\limits_{n=0}^\infty x^nq^{\binom n2}$, considered as a formal power series. He proved that all the coefficients of $$-x_0(q)=1+q+2q^2+4q^3+9q^4+...$$ are positive integers. I give here an explicit combinatorial interpretation of these coefficients. More precisely, I show that $-x_0(q)$ enumerates rooted trees that are enriched by certain polyominoes, weighted according to their total area.Comment: 15 pages, 7 figure

Winding angles for two-dimensional polymers with orientation dependent interactions

We study winding angles of oriented polymers with orientation-dependent interaction in two dimensions. Using exact analytical calculations, computer simulations, and phenomenological arguments, we succeed in finding the variance of the winding angle for most of the phase diagram. Our results suggest that the winding angle distribution is a universal quantity, and that the $\theta$--point is the point where the three phase boundaries between the swollen, the normal collapsed, and the spiral collapsed phase meet. The transition between the normal collapsed phase and the spiral phase is shown to be continuous.Comment: 21 pages (incl 5 figures

Dynamics of a single particle in a horizontally shaken box

We study the dynamics of a particle in a horizontally and periodically shaken box as a function of the box parameters and the coefficient of restitution. For certain parameter values, the particle becomes regularly chattered at one of the walls, thereby loosing all its kinetic energy relative to that wall. The number of container oscillations between two chattering events depends in a fractal manner on the parameters of the system. In contrast to a vertically vibrated particle, for which chattering is claimed to be the generic fate, the horizontally shaken particle can become trapped on a periodic orbit and follow the period-doubling route to chaos when the coefficient of restitution is changed. We also discuss the case of a completely elastic particle, and the influence of friction between the particle and the bottom of the container.Comment: 11 pages RevTex. Some postscript files have low resolution. We will send the high-resolution files on reques

Uniform asymptotics of area-weighted Dyck paths

Using the generalized method of steepest descents for the case of two coalescing saddle points, we derive an asymptotic expression for the bivariate generating function of Dyck paths, weighted according to their length and their area in the limit of the area generating variable tending towards 1. The result is valid uniformly for a range of the length generating variable, including the tricritical point of the model.Comment: 14 pages, 5 figure

Pressure exerted by a vesicle on a surface

Several recent works have considered the pressure exerted on a wall by a model polymer. We extend this consideration to vesicles attached to a wall, and hence include osmotic pressure. We do this by considering a two-dimensional directed model, namely that of area-weighted Dyck paths. Not surprisingly, the pressure exerted by the vesicle on the wall depends on the osmotic pressure inside, especially its sign. Here, we discuss the scaling of this pressure in the different regimes, paying particular attention to the crossover between positive and negative osmotic pressure. In our directed model, there exists an underlying Airy function scaling form, from which we extract the dependence of the bulk pressure on small osmotic pressures.Comment: 10 pages, 7 figure