Related papers: Self-Avoiding Walks with Writhe

For $d \geq 2$ and $n \in \mathbb{N}$, let $\mathsf{W}_n$ denote the uniform law on self-avoiding walks beginning at the origin in the integer lattice $\mathbb{Z}^d$, and write $\Gamma$ for a $\mathsf{W}_n$-distributed walk. We show that…

We consider the random walk of a particle in a two-dimensional self-affine random potential of Hurst exponent $H=1/2$ in the presence of an external force $F$. We present numerical results on the statistics of first-passage times that…

Several kinds of walks on complex networks are currently used to analyze search and navigation in different systems. Many analytical and computational results are known for random walks on such networks. Self-avoiding walks (SAWs) are…

Topological properties of crystalline ice structures are studied by means of self-avoiding walks on their H-bond networks. The number of self-avoiding walks, C_n, for eight ice polymorphs has been obtained by direct enumeration up to walk…

The Hammersley-Welsh bound (1962) states that the number $c_n$ of length $n$ self-avoiding walks on $\mathbb{Z}^d$ satisfies \[ c_n \leq \exp \left[ O(n^{1/2}) \right] \mu_c^n, \] where $\mu_c=\mu_c(d)$ is the connective constant of…

The discrete quantum walk in N dimensions is analyzed from the perspective of its dispersion relations. This allows understanding known properties, as well as designing new ones when spatially extended initial conditions are considered.…

We study the effective radius of weakly self-avoiding star polymers in one, two, and three dimensions. Our model includes $N$ Brownian motions up to time $T$, started at the origin and subject to exponential penalization based on the amount…

For $d \geq 2$ and $n \in \mathbb{N}$, let $\mathsf{W}_n$ denote the uniform law on self-avoiding walks of length $n$ beginning at the origin in the nearest-neighbour integer lattice $\mathbb{Z}^d$, and write $\Gamma$ for a…

Recently it has been argued that weighting the writhe of unknotted self-avoiding polygons can be related to possible experiments that turn double stranded DNA. We first solve exactly a directed model and demonstrate that in such a subset of…

We consider two dimensional random walks conditioned to stay in the positive quadrant. Assuming that the increments of the walk have finite second moments and that the drift vector is co-oriented with one of two axes, we construct positive…

This work is motivated by the study of some two-dimensional random walks in random environment (RWRE) with transition probabilities independent of one coordinate of the walk. These are non-reversible models and can not be treated by…

We study the possibility for branching random walks in random environment (BRWRE) to survive. The particles perform simple symmetric random walks on the $d$-dimensional integer lattice, while at each time unit, they split into independent…

Consider a centred random walk in dimension one with a positive finite variance $\sigma^2$, and let $\tau_B$ be the hitting time for a bounded Borel set $B$ with a non-empty interior. We prove the asymptotic $P_x(\tau_B > n) \sim \sqrt{2 /…

We give explicitly in the closed formulae the genus zero primary potentials of the three 6-dimensional FJRW theories of the simple-elliptic singularity $\tilde E_7$ with the non-maximal symmetry groups. For each of these FJRW theories we…

These lecture notes provide a rapid introduction to a number of rigorous results on self-avoiding walks, with emphasis on the critical behaviour. Following an introductory overview of the central problems, an account is given of the…

A planar self-avoiding walk (SAW) is a nearest neighbor random walk path in the square lattice with no self-intersection. A planar self-avoiding polygon (SAP) is a loop with no self-intersection. In this paper we present conjectures for the…

We theoretically study the self-propulsion of a thin (slender) colloid driven by asymmetric chemical reactions on its surface at vanishing Reynolds number. Using the method of matched asymptotic expansions, we obtain the colloid…

A new solution in the string background dual to N=1 SQCD-like theories is presented. The gauge coupling in this solution has walking property. The Wilson loop calculations show that quark anti-quark potential makes phase transitions.…

We study the winding angles of random and self-avoiding walks on square and cubic lattices with number of steps $N$ ranging up to $10^7$. We show that the mean square winding angle $\langle\theta^2\rangle$ of random walks converges to the…

We study self-avoiding walks on three-dimensional critical percolation clusters using a new exact enumeration method. It overcomes the exponential increase in computation time by exploiting the clusters' fractal nature. We enumerate walks…