arXiv subjects

Kilian Raschel

Publications and source records attributed to Kilian Raschel.

At least 55 records · Page 3Linked to original sources

arXiv2020

Polyharmonic functions and random processes in cones

We investigate polyharmonic functions associated to Brownian motion and random walks in cones. These are functions which cancel some power of the usual Laplacian in the continuous setting and of the discrete Laplacian in the discrete setting. We show that polyharmonic functions naturally appear while considering asymptotic expansions of the heat kernel in the Brownian case and in lattice walk enumeration problems. We provide a method to construct general polyharmonic functions through Laplace transforms and generating functions in the continuous and discrete cases, respectively. This is done by using a functional equation approach.

math.CO↗

arXiv2020

Martin boundary of random walks in convex cones

We determine the asymptotic behavior of the Green function for zero-drift random walks confined to multidimensional convex cones. As a consequence, we prove that there is a unique positive discrete harmonic function for these processes (up to a multiplicative constant); in other words, the Martin boundary reduces to a singleton.

math.PR↗

arXiv2020

Boundary behavior for random walks in cones

We study the asymptotic behavior of zero-drift random walks confined to multidimensional convex cones, when the endpoint is close to the boundary. We derive a local limit theorem in the fluctuation regime.

math.PR↗

arXiv2019

On the least common multiple of several random integers

Let $L_n(k)$ denote the least common multiple of $k$ independent random integers uniformly chosen in $\{1,2,\ldots ,n\}$. In this note, using a purely probabilistic approach, we derive a criterion for the convergence in distribution as $n\to\infty$ of $\frac{f(L_n(k))}{n^{rk}}$ for a wide class of multiplicative arithmetic functions~$f$ with polynomial growth $r>-1$. Furthermore, we identify the limit as an infinite product of independent random variables indexed by prime numbers. Along the way, we compute the generating function of a trimmed sum of independent geometric laws, occurring in the above infinite product. This generating function is rational; we relate it to the generating function of a certain max-type Diophantine equation, of which we solve a generalized version. Our results extend theorems by Erdős and Wintner (1939), Fernández and Fernández (2013) and Hilberdink and Tóth (2016).

math.PR↗

arXiv2019

The extinction problem for a class of distylous plant populations

In this paper, the extinction problem for a class of distylous plant populations is considered within the framework of certain nonhomogeneous nearest-neighbor random walks in the positive quadrant. For the latter, extinction means absorption at one of the axes. Despite connections with some classical probabilistic models (standard two-type Galton-Watson process, two-urn model), exact formulae for the probabilities of absorption seem to be difficult to come by and one must therefore resort to good approximations. In order to meet this task, we develop potential-theoretic tools and provide various sub- and super-harmonic functions which, for large initial populations, provide bounds which in particular improve those that have appeared earlier in the literature.

math.PR↗

arXiv2019

On walks avoiding a quadrant

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 the first quadrant have been studied a lot, the case of planar, non-convex cones---equivalent to the three-quarter plane after a linear transform---has been approached only recently. In this article we develop an analytic approach to the case of walks in three quadrants. The advantage of this method is to provide uniform treatment in the study of models corresponding to different step sets. After splitting the three quadrants in two symmetric convex cones, the method is composed of three main steps: write a system of functional equations satisfied by the counting generating function, which may be simplified into one single equation under symmetry conditions; transform the functional equation into a boundary value problem; and finally solve this problem, using a concept of anti-Tutte's invariant. The result is a contour-integral expression for the generating function. Such systems of functional equations also appear in queueing theory with the famous Join-the-Shortest-Queue model, which is still an open problem in the non-symmetric case.

math.CO↗

arXiv2018

Differential transcendence & algebraicity criteria for the series counting weighted quadrant walks

We consider weighted small step walks in the positive quadrant, and provide algebraicity and differential transcendence results for the underlying generating functions: we prove that depending on the probabilities of allowed steps, certain of the generating functions are algebraic over the field of rational functions, while some others do not satisfy any algebraic differential equation with rational function coefficients. Our techniques involve differential Galois theory for difference equations as well as complex analysis (Weierstrass parameterization of elliptic curves). We also extend to the weighted case many key intermediate results, as a theorem of analytic continuation of the generating functions.

math.CO↗

arXiv2017

Tutte's invariant approach for Brownian motion reflected in the quadrant

We consider a Brownian motion with drift in the quarter plane with orthogonal reflection on the axes. The Laplace transform of its stationary distribution satisfies a functional equation, which is reminiscent from equations arising in the enumeration of (discrete) quadrant walks. We develop a Tutte's invariant approach to this continuous setting, and we obtain an explicit formula for the Laplace transform in terms of generalized Chebyshev polynomials.

math.PR↗

arXiv2017

Weighted Lattice Walks and Universality Classes

In this work we consider two different aspects of weighted walks in cones. To begin we examine a particular weighted model, known as the Gouyou-Beauchamps model. Using the theory of analytic combinatorics in several variables we obtain the asymptotic expansion of the total number of Gouyou-Beauchamps walks confined to the quarter plane. Our formulas are parametrized by weights and starting point, and we identify six different asymptotic regimes (called universality classes) which arise according to the values of the weights. The weights allowed in this model satisfy natural algebraic identities permitting an expression of the weighted generating function in terms of the generating function of unweighted walks on the same steps. The second part of this article explains these identities combinatorially for walks in arbitrary cones and dimensions, and provides a characterization of universality classes for general weighted walks. Furthermore, we describe an infinite set of models with non-D-finite generating function.

math.CO↗

arXiv2017

The $Z$-invariant massive Laplacian on isoradial graphs

We introduce a one-parameter family of massive Laplacian operators $(Δ^{m(k)})_{k\in[0,1)}$ defined on isoradial graphs, involving elliptic functions. We prove an explicit formula for the inverse of $Δ^{m(k)}$, the massive Green function, which has the remarkable property of only depending on the local geometry of the graph, and compute its asymptotics. We study the corresponding statistical mechanics model of random rooted spanning forests. We prove an explicit local formula for an infinite volume Boltzmann measure, and for the free energy of the model. We show that the model undergoes a second order phase transition at $k=0$, thus proving that spanning trees corresponding to the Laplacian introduced by Kenyon are critical. We prove that the massive Laplacian operators $(Δ^{m(k)})_{k\in(0,1)}$ provide a one-parameter family of $Z$-invariant rooted spanning forest models. When the isoradial graph is moreover $\mathbb{Z}^2$-periodic, we consider the spectral curve of the characteristic polynomial of the massive Laplacian. We provide an explicit parametrization of the curve and prove that it is Harnack and has genus $1$. We further show that every Harnack curve of genus $1$ with $(z,w)\leftrightarrow(z^{-1},w^{-1})$ symmetry arises from such a massive Laplacian.

math.PR↗

arXiv2016

Analytic approach for reflected Brownian motion in the quadrant

Random walks in the quarter plane are an important object both of combinatorics and probability theory. Of particular interest for their study, there is an analytic approach initiated by Fayolle, Iasnogorodski and Malyshev, and further developed by the last two authors of this note. The outcomes of this method are explicit expressions for the generating functions of interest, asymptotic analysis of their coefficients, etc. Although there is an important literature on reflected Brownian motion in the quarter plane (the continuous counterpart of quadrant random walks), an analogue of the analytic approach has not been fully developed to that context. The aim of this note is twofold: it is first an extended abstract of two recent articles of the authors of this paper, which propose such an approach; we further compare various aspects of the discrete and continuous analytic approaches.

math.PR↗

arXiv2016

Counting quadrant walks via Tutte's invariant method (extended abstract)

In the 1970s, Tutte developed a clever algebraic approach, based on certain "invariants" , to solve a functional equation that arises in the enumeration of properly colored triangulations. The enumeration of plane lattice walks confined to the first quadrant is governed by similar equations, and has led in the past decade to a rich collection of attractive results dealing with the nature (algebraic, D-finite or not) of the associated generating function, depending on the set of allowed steps. We first adapt Tutte's approach to prove (or reprove) the algebraicity of all quadrant models known or conjectured to be algebraic (with one small exception). This includes Gessel's famous model, and the first proof ever found for one model with weighted steps. To be applicable, the method requires the existence of two rational functions called invariant and decoupling function respectively. When they exist, algebraicity comes out (almost) automatically. Then, we move to an analytic viewpoint which has already proved very powerful, leading in particular to integral expressions of the generating function in the non-D-finite cases, as well as to proofs of non-D-finiteness. We develop in this context a weaker notion of invariant. Now all quadrant models have invariants, and for those that have in addition a decoupling function, we obtain integral-free expressions of the generating function, and a proof that this series is differentially algebraic (that is, satisfies a non-linear differential equation).

math.CO↗

arXiv2016

$t$-Martin boundary of killed random walks in the quadrant

We compute the $t$-Martin boundary of two-dimensional small steps random walks killed at the boundary of the quarter plane. We further provide explicit expressions for the (generating functions of the) discrete $t$-harmonic functions. Our approach is uniform in $t$, and shows that there are three regimes for the Martin boundary.

math.PR↗

arXiv2015

On the exit time from a cone for random walks with drift

We compute the exponential decay of the probability that a given multi-dimensional random walk stays in a convex cone up to time $n$, as $n$ goes to infinity. We show that the latter equals the minimum, on the dual cone, of the Laplace transform of the random walk increments. As an example, our results find applications in the counting of walks in orthants, a classical domain in enumerative combinatorics.

math.PR↗

arXiv2015

A human proof of Gessel's lattice path conjecture

Gessel walks are lattice paths confined to the quarter plane that start at the origin and consist of unit steps going either West, East, South-West or North-East. In 2001, Ira Gessel conjectured a nice closed-form expression for the number of Gessel walks ending at the origin. In 2008, Kauers, Koutschan and Zeilberger gave a computer-aided proof of this conjecture. The same year, Bostan and Kauers showed, again using computer algebra tools, that the complete generating function of Gessel walks is algebraic. In this article we propose the first "human proofs" of these results. They are derived from a new expression for the generating function of Gessel walks in terms of Weierstrass zeta functions.

math.CO↗

arXiv2015

New steps in walks with small steps in the quarter plane

In this article we obtain new expressions for the generating functions counting (non-singular) walks with small steps in the quarter plane. Those are given in terms of infinite series, while in the literature, the standard expressions use solutions to boundary value problems. We illustrate our results with three examples (an algebraic case, a transcendental D-finite case, and an infinite group model).

math.CO↗

arXiv2015

About a possible analytic approach for walks in the quarter plane with arbitrary big jumps

In this note, we consider random walks in the quarter plane with arbitrary big jumps. We announce the extension to that class of models of the analytic approach of [G. Fayolle, R. Iasnogorodski, and V. Malyshev, Random walks in the quarter plane, Springer-Verlag, Berlin (1999)], initially valid for walks with small steps in the quarter plane. New technical challenges arise, most of them being tackled in the framework of generalized boundary value problems on compact Riemann surfaces.

math.PR↗

arXiv2014

On the exit time from a cone for Brownian motion with drift

We investigate the tail distribution of the first exit time of Brownian motion with drift from a cone and find its exact asymptotics for a large class of cones. Our results show in particular that its exponential decreasing rate is a function of the distance between the drift and the cone, whereas the polynomial part in the asymptotics depends on the position of the drift with respect to the cone and its polar cone, and reflects the local geometry of the cone at the point where the drift is orthogonally projected.

math.PR↗