Related papers: Weyl chambers for short step Quarter-plane Lattice…
In this article, we put forward a practical but generic approach towards constructing a large family of $(3+1)$ dimension lattice models which can naturally lead to a single Weyl cone in the infrared (IR) limit. Our proposal relies on…
In the past decade, a lot of attention has been devoted to the enumera-tion of walks with prescribed steps confined to a convex cone. In two dimensions, this means counting walks in the first quadrant of the plane (possibly after a linear…
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…
The Weber problem consists of finding a point in $\mathbbm{R}^n$ that minimizes the weighted sum of distances from $m$ points in $\mathbbm{R}^n$ that are not collinear. An application that motivated this problem is the optimal location of…
Recently, several authors have considered lattice paths with various steps, including vertical steps permitted. In this paper, we consider a kind of generalized Motzkin paths, called {\it G-Motzkin paths} for short, that is lattice paths…
We show that every Weyl module for a current algebra has a filtration whose successive quotients are isomorphic to Demazure modules, and that the path model for a tensor product of level zero fundamental representations is isomorphic to a…
The Heisenberg-Weyl algebra, which underlies virtually all physical representations of Quantum Theory, is considered from the combinatorial point of view. We provide a concrete model of the algebra in terms of paths on a lattice with some…
Inspired by a new mathematical model for bobbin lace, this paper considers finite lattice paths formed from the set of step vectors $\mathfrak{A}=$$\{\rightarrow,$ $\nearrow,$ $\searrow,$ $\uparrow,$ $\downarrow\}$ with the restriction that…
In the time-reversal-breaking centrosymmetric systems, the appearance of Weyl points can be guaranteed by an odd number of all the even/odd parity occupied bands at eight inversion-symmetry-invariant momenta. Here, based on symmetry…
We count a large class of lattice paths by using factorizations of free monoids. Besides the classical lattice paths counting problems related to Catalan numbers, we give a new approach to the problem of counting walks on the slit plane…
Quadratic matrix equations of the kind $A_1X^2+A_0X+A_{-1}=X$ are encountered in the analysis of Quasi--Birth-Death stochastic processes where the solution of interest is the minimal nonnegative solution $G$. In many queueing models,…
In the past 20 years, the enumeration of plane lattice walks confined to a convex cone -- normalized into the first quadrant -- has received a lot of attention, stimulated the development of several original approaches, and led to a rich…
Enumeration of planar lattice walks is a classical topic in combinatorics, at the cross-roads of several domains (e.g., probability, statistical physics, computer science). The aim of this paper is to propose a new approach to obtain some…
In this article, we study the enumeration by length of several walk models on the square lattice. We obtain bijections between walks in the upper half-plane returning to the $x$-axis and walks in the quarter plane. A recent work by Bostan,…
It is well known that the set of $m$-Dyck paths with a fixed height and a fixed amount of valleys is counted by the Fu{\ss}-Narayana numbers. In this article, we consider the set of $m$-Dyck paths that start with at least $t$ north steps.…
We present an algorithm, based on the iteration of conformal maps, that produces independent samples of self-avoiding paths in the plane. It is a discrete process approximating radial Schramm-Loewner evolution growing to infinity. We focus…
We study the topological dynamics of the action of the diagonal subgroup on quotients Gamma\PSL(2,R)*PSL(2,R), where Gamma is an irreducible lattice. Closed orbits are described and a set of points of dense orbit is explicitly given. Such…
We present a feasible protocol to mimic topological Weyl semimetal phase in a small one-dimensional circuit-QED lattice. By modulating the photon hopping rates and on-site photon frequencies in parametric spaces, we demonstrate that the…
Let S be a finite subset of Z^2. A walk on the slit plane with steps in S is a sequence (0,0)=w_0, w_1, ..., w_n of points of Z^2 such that w_{i+1}-w_i belongs to S for all i, and none of the points w_i, i>0, lie on the half-line H= {(k,0):…
We define a natural lattice structure on all subsets of a finite root system that extends the weak order on the elements of the corresponding Coxeter group. For crystallographic root systems, we show that the subposet of this lattice…