Related papers: Towards a Combinatorial Understanding of Lattice P…

We consider a discrete random walk on a diagonal lattice in two and three dimensions and obtain explicit solutions of absorption probabilities and probabilities of return in several domains. In three dimensions we consider both the cube and…

An important problem in analytic and geometric combinatorics is estimating the number of lattice points in a compact convex set in a Euclidean space. Such estimates have numerous applications throughout mathematics. In this note, we exhibit…

This paper extends the inductive approach to the lace expansion of van der Hofstad and Slade in order to prove Gaussian asymptotic behaviour for models with critical dimension other than 4. The results are applied by Holmes to study…

We discuss spreading estimates for dynamical systems given by the iteration of an extended CMV matrix. Using a connection due to Cantero--Gr\"unbaum--Moral--Vel\'azquez, this enables us to study spreading rates for quantum walks in one…

We consider a continuous-time branching random walk on a multidimensional lattice in a random branching medium. It is theoretically known that, in such branching random walks, large rare fluctuations of the medium may lead to anomalous…

Two-dimensional (random) walks in cones are very natural both in combinatorics and probability theory: they are interesting for themselves and also because they are strongly related to other discrete structures. While walks restricted to…

This work is devoted to the analysis of a Gibbs partition model, also known as a composition scheme. We consider a natural new condition on the component weights. It leads to a new behavior for the total number of components. We discover a…

The Riemann walk is the lattice version of the Levy flight. For the one-dimensional Riemann walk of Levy exponent 0<\alpha<2 we study the statistics of the support, i.e. the set of visited sites, after t steps. We consider a wide class of…

We infer upper and lower bounds on the exponential growth constants $\alpha(\Lambda)$, $\alpha_0(\Lambda)$, and $\beta(\Lambda)$ describing the large-$n$ behavior of, respectively, the number of acyclic orientations, acyclic orientations…

We consider a random walk on the first quadrant of the square lattice, whose increment law is, roughly speaking, homogeneous along a finite number of half-lines near each of the two boundaries, and hence essentially specified by…

Relations between the mean values of distributions of flipped spins on periodic Heisenberg XX chain and some aspects of enumerative combinatorics are discussed. The Bethe vectors, which are the state-vectors of the model, are considered…

In probability theory, reinforced walks are random walks on a lattice (or more generally a graph) that preferentially revisit neighboring `locations' (sites or bonds) that have been visited before. In this paper, we consider walks with…

The enumeration of quarter-plane lattice walks with small steps is a classical problem in combinatorics. An effective approach is the kernel method, where the solution is derived by positive term extraction. Alternatively, one may reduce…

This paper investigates L\'evy walks with random velocities, extending classical models beyond constant speed assumptions. We derive scaling limits, demonstrating that diffusion depends on interplay between heavy-tailed duration and…

We consider uniform random permutations drawn from a family enumerated through generating trees. We develop a new general technique to establish a central limit theorem for the number of consecutive occurrences of a fixed pattern in such…

The main theme of this dissertation is retooling methods to work for different situations. I have taken the method derived by O'Hara and simplified by Zeilberger to prove unimodality of $q$-binomials and tweaked it. This allows us to create…

We derive a path counting formula for two-dimensional lattice path model with filter restrictions in the presence of long steps, source and target points of which are situated near the filters. This solves a problem of finding an explicit…

We consider a class of stochastic growth models on the integer lattice which includes various interesting examples such as the number of open paths in oriented percolation and the binary contact path process. Under some mild assumptions, we…

In the present paper, we use difference Galois theory to study the nature of the generating function counting walks with small steps in the quarter plane. These series are trivariate formal power series $Q(x,y,t)$ that count the number of…

We consider the dynamical properties of Quantum Walks defined on the d-dimensional cubic lattice, or the homogeneous tree of coordination number 2d, with site dependent random phases, further characterised by transition probabilities…