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Scaling limit and cube-root fluctuations in SOS surfaces above a wall

Pietro Caputo, Eyal Lubetzky, Fabio Martinelli, Allan Sly, Fabio Lucio Toninelli (2016)

Journal of the European Mathematical Society

Consider the classical $(2 + 1)$ -dimensional Solid-On-Solid model above a hard wall on an $L \times L$ box of $ℤ^{2}$ . The model describes a crystal surface by assigning a non-negative integer height $η_{x}$ to each site $x$ in the box and 0 heights to its boundary. The probability of a surface configuration $η$ is proportional to $\exp (- β ℋ (η))$ , where $β$ is the inverse-temperature and $ℋ (η)$ sums the absolute values of height differences between neighboring sites. We give a full description of the shape of the SOS surface for low enough temperatures....

Scaling of a random walk on a supercritical contact process

F. den Hollander, R. S. dos Santos (2014)

Annales de l'I.H.P. Probabilités et statistiques

We prove a strong law of large numbers for a one-dimensional random walk in a dynamic random environment given by a supercritical contact process in equilibrium. The proof uses a coupling argument based on the observation that the random walk eventually gets trapped inside the union of space–time cones contained in the infection clusters generated by single infections. In the case where the local drifts of the random walk are smaller than the speed at which infection clusters grow, the random walk...

Sectorial local non-determinism and the geometry of the Brownian sheet.

Xiao, Yimin, Khoshnevisan, Davar, Wu, Dongsheng (2006)

Electronic Journal of Probability [electronic only]

Self-avoiding walks on the lattice ℤ² with the 8-neighbourhood system

Andrzej Chydziński, Bogdan Smołka (2001)

Applicationes Mathematicae

This paper deals with the properties of self-avoiding walks defined on the lattice with the 8-neighbourhood system. We compute the number of walks, bridges and mean-square displacement for N=1 through 13 (N is the number of steps of the self-avoiding walk). We also estimate the connective constant and critical exponents, and study finite memory and generating functions. We show applications of this kind of walk. In addition, we compute upper bounds for the number of walks and the connective constant....

Self-repelling random walk with directed edges on Z.

Veto, Balint, Tóth, Bálint (2008)

Electronic Journal of Probability [electronic only]

Strong disorder in semidirected random polymers

N. Zygouras (2013)

Annales de l'I.H.P. Probabilités et statistiques

We consider a random walk in a random potential, which models a situation of a random polymer and we study the annealed and quenched costs to perform long crossings from a point to a hyperplane. These costs are measured by the so called Lyapounov norms. We identify situations where the point-to-hyperplane annealed and quenched Lyapounov norms are different. We also prove that in these cases the polymer path exhibits localization.

Supersymmetry, Witten complex and asymptotics for directional Lyapunov exponents in $𝐙^{d}$

Wei-Min Wang (1999)

Journées équations aux dérivées partielles

By using a supersymmetric gaussian representation, we transform the averaged Green's function for random walks in random potentials into a 2-point correlation function of a corresponding lattice field theory. We study the resulting lattice field theory using the Witten laplacian formulation. We obtain the asymptotics for the directional Lyapunov exponents.

Survival time of random walk in random environment among soft obstacles.

Gantert, Nina, Popov, Serguei, Vachkovskaia, Marina (2009)

Electronic Journal of Probability [electronic only]

Symbol-crunching with the transfer-matrix method in order to count skinny physical creatures.

Zeilberger, Doron (2000)

Integers

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