Related papers: The generating function of Kreweras walks with int…

Prudent walks are special self-avoiding walks that never take a step towards an already occupied site, and \emph{$k$-sided prudent walks} (with $k=1,2,3,4$) are, in essence, only allowed to grow along $k$ directions. Prudent polygons are…

We prove that the anisotropic generating function of self-avoiding polygons is not a D-finite function - proving a conjecture of Guttmann and Enting. This result is also generalised to self-avoiding polygons on hypercubic lattices. Using…

Let V be an infinite matrix with rows and columns indexed by the positive integers, and entries in a field F. Suppose that v_{i,j} only depends on i-j and is 0 for |i-j| large. Then V^n is defined for all n, and one has a "generating…

We construct harmonic functions in the quarter plane for discrete Laplace operators. In particular, the functions are conditioned to vanish on the boundary and the Laplacians admit coefficients associated with transition probabilities of…

We consider a two dimensional reflecting random walk on the nonnegative integer quadrant. It is assumed that this reflecting random walk has skip free transitions. We are concerned with its time reversed process assuming that the stationary…

We investigate reflected random walks in the quarter plane, with particular emphasis on the time spent along the reflection boundary axes. Assuming the drift of the random walk lies within the cone, the local time converges -- without the…

Let W be a finite Coxeter group. We define its Hecke-group algebra by gluing together appropriately its group algebra and its 0-Hecke algebra. We describe in detail this algebra (dimension, several bases, conjectural presentation,…

Prudent walks are self-avoiding walks on the square lattice which never step into the direction of an already occupied vertex. We study the closed version of these walks, called prudent polygons, where the last vertex is adjacent to the…

We describe a generating tree approach to the enumeration and exhaustive generation of k-nonnesting set partitions and permutations. Unlike previous work in the literature using the connections of these objects to Young tableaux and…

We study a class of algebras with non-Lie commutation relations whose symplectic leaves are surfaces of revolution: a cylinder or a torus. Over each of such surfaces we introduce a family of complex structures and Hilbert spaces of…

Recently, Bostan and his coauthors investigated lattice walks restricted to the non-negative octant $\mathbb{N}^3$. For the $35548$ non-trivial models with at most six steps, they found that many models associated to a group of order at…

Consider a projective limit G of finite groups G_n. Fix a compatible family \delta^n of coactions of the G_n on a C*-algebra A. From this data we obtain a coaction \delta of G on A. We show that the coaction crossed product of A by \delta…

Based on studies on four specific networks, we conjecture a general relation between the walk dimensions $d_{w}$ of discrete-time random walks and quantum walks with the (self-inverse) Grover coin. In each case, we find that $d_{w}$ of the…

We extend the classification of nearest neighbour walks in the quarter plane to models in which multiplicities are attached to each direction in the step set. Our study leads to a small number of infinite families that completely…

We consider an abstract quantum walk defined by a unitary evolution operator $U$, which acts on a Hilbert space decomposed into a direct sum of Hilbert spaces $\{\mathcal{H}_v \}_{v \in V}$. We show that such $U$ naturally defines a…

Ideas of Kn\"odel and B\"ohm-Hornik about walks in certain graphs, resembling the classical symmetric random walk on the integers, are combined. All the relevant generating functions (although occasionally quite involved) are made fully…

In the context of lattice walk enumeration in cones, we consider the number of walks in the quarter plane with fixed starting and ending points, prescribed step-set and given length. After renormalization, this number may be interpreted as…

Every symmetric generating functional of a convolution semigroup of states on a locally compact quantum group is shown to admit a dense unital $*$-subalgebra with core-like properties in its domain. On the other hand we prove that every…

Sequences are often conveniently encoded in the form of a generating function depending on a formal variable. This note presents two observations that allow one to draw conclusions about the generated sequence from the generating function.…

Consider a one dimensional simple random walk $X=(X_n)_{n\geq0}$. We form a new simple symmetric random walk $Y=(Y_n)_{n\geq0}$ by taking sums of products of the increments of $X$ and study the two-dimensional walk…