Nonequilibrium stationary states and equilibrium models with long range interactions

It was recently suggested by Blythe and Evans that a properly defined steady state normalisation factor can be seen as a partition function of a fictitious statistical ensemble in which the transition rates of the stochastic process play the role of fugacities. In analogy with the Lee-Yang description of phase transition of equilibrium systems, they studied the zeroes in the complex plane of the normalisation factor in order to find phase transitions in nonequilibrium steady states. We show that like for equilibrium systems, the ``densities'' associated to the rates are non-decreasing functions of the rates and therefore one can obtain the location and nature of phase transitions directly from the analytical properties of the ``densities''. We illustrate this phenomenon for the asymmetric exclusion process. We actually show that its normalisation factor coincides with an equilibrium partition function of a walk model in which the ``densities'' have a simple physical interpretation.Comment: LaTeX, 23 pages, 3 EPS figure

Scaling function and universal amplitude combinations for self-avoiding polygons

We analyze new data for self-avoiding polygons, on the square and triangular lattices, enumerated by both perimeter and area, providing evidence that the scaling function is the logarithm of an Airy function. The results imply universal amplitude combinations for all area moments and suggest that rooted self-avoiding polygons may satisfy a $q$-algebraic functional equation.Comment: 9 page

Asymptotic behaviour of convex and column-convex lattice polygons with fixed area and varying perimeter

We study the inflated phase of two dimensional lattice polygons, both convex and column-convex, with fixed area A and variable perimeter, when a weight \mu^t \exp[- Jb] is associated to a polygon with perimeter t and b bends. The mean perimeter is calculated as a function of the fugacity \mu and the bending rigidity J. In the limit \mu -> 0, the mean perimeter has the asymptotic behaviour \avg{t}/4 \sqrt{A} \simeq 1 - K(J)/(\ln \mu)^2 + O (\mu/ \ln \mu) . The constant K(J) is found to be the same for both types of polygons, suggesting that self-avoiding polygons should also exhibit the same asymptotic behaviour.Comment: 10 pages, 3 figure

Staircase polygons: moments of diagonal lengths and column heights

We consider staircase polygons, counted by perimeter and sums of k-th powers of their diagonal lengths, k being a positive integer. We derive limit distributions for these parameters in the limit of large perimeter and compare the results to Monte-Carlo simulations of self-avoiding polygons. We also analyse staircase polygons, counted by width and sums of powers of their column heights, and we apply our methods to related models of directed walks.Comment: 24 pages, 7 figures; to appear in proceedings of Counting Complexity: An International Workshop On Statistical Mechanics And Combinatorics, 10-15 July 2005, Queensland, Australi

Exact Scaling Functions for Self-Avoiding Loops and Branched Polymers

It is shown that a recently conjectured form for the critical scaling function for planar self-avoiding polygons weighted by their perimeter and area also follows from an exact renormalization group flow into the branched polymer problem, combined with the dimensional reduction arguments of Parisi and Sourlas. The result is generalized to higher-order multicritical points, yielding exact values for all their critical exponents and exact forms for the associated scaling functions.Comment: 5 pages; v2: factors of 2 corrected; v.3: relation with existing theta-point results clarified, some references added/update

Asymptotic Behavior of Inflated Lattice Polygons

We study the inflated phase of two dimensional lattice polygons with fixed perimeter $N$ and variable area, associating a weight $\exp[pA - Jb ]$ to a polygon with area $A$ and $b$ bends. For convex and column-convex polygons, we show that $/A_{max} = 1 - K(J)/\tilde{p}^2 + \mathcal{O}(\rho^{-\tilde{p}})$, where $\tilde{p}=pN \gg 1$, and $\rho<1$. The constant $K(J)$ is found to be the same for both types of polygons. We argue that self-avoiding polygons should exhibit the same asymptotic behavior. For self-avoiding polygons, our predictions are in good agreement with exact enumeration data for J=0 and Monte Carlo simulations for $J \neq 0$. We also study polygons where self-intersections are allowed, verifying numerically that the asymptotic behavior described above continues to hold.Comment: 7 page