arXiv subjects

Kilian Raschel

Publications and source records attributed to Kilian Raschel.

At least 37 records · Page 2Linked to original sources

arXiv2023

Random Walks in the High-Dimensional Limit II: The Crinkled Subordinator

A crinkled subordinator is an $\ell^2$-valued random process which can be thought of as a version of the usual one-dimensional subordinator with each out of countably many jumps being in a direction orthogonal to the directions of all other jumps. We show that the path of a $d$-dimensional random walk with $n$ independent identically distributed steps with heavy-tailed distribution of the radial components and asymptotically orthogonal angular components converges in distribution in the Hausdorff distance up to isometry and also in the Gromov--Hausdorff sense, if viewed as a random metric space, to the closed range of a crinkled subordinator, as $d,n\to\infty$.

math.PR↗

arXiv2023

Multivariate multiplicative functions of uniform random vectors in large integer domains

For a wide class of sequences of integer domains $\mathcal{D}_n\subset\mathbb{N}^d$, $n\in\mathbb{N}$, we prove distributional limit theorems for $F(X_1^{(n)},\ldots,X_d^{(n)})$, where $F$ is a multivariate multiplicative function and $(X_1^{(n)},\ldots,X_d^{(n)})$ is a random vector with uniform distribution on $\mathcal{D}_n$. As a corollary, we obtain limit theorems for the greatest common divisor and least common multiple of the random set $\{X_1^{(n)},\ldots,X_d^{(n)}\}$. This generalizes previously known limit results for $\mathcal{D}_n$ being either a discrete cube or a discrete hyperbolic region.

math.PR↗

arXiv2023

Constructing discrete harmonic functions in wedges

We propose a systematic construction of signed harmonic functions for discrete Laplacian operators with Dirichlet conditions in the quarter plane. In particular, we prove that the set of harmonic functions is an algebra generated by a single element, which conjecturally corresponds to the unique positive harmonic function.

math.SP↗

arXiv2022

On the stationary distribution of reflected Brownian motion in a non-convex wedge

We study the stationary reflected Brownian motion in a non-convex wedge, which, compared to its convex analogue model, has been much rarely analyzed in the probabilistic literature. We prove that its stationary distribution can be found by solving a two dimensional vector boundary value problem (BVP) on a single curve for the associated Laplace transforms. The reduction to this kind of vector BVP seems to be new in the literature. As a matter of comparison, one single boundary condition is sufficient in the convex case. When the parameters of the model (drift, reflection angles and covariance matrix) are symmetric with respect to the bisector line of the cone, the model is reducible to a standard reflected Brownian motion in a convex cone. Finally, we construct a one-parameter family of distributions, which surprisingly provides, for any wedge (convex or not), one particular example of stationary distribution of a reflected Brownian motion.

math.PR↗

arXiv2022

Marches aléatoires dans un cône et fonctions discrètes harmoniques

Random walks in cones have the double interest of being at the heart of many probabilistic problems and of being related to many mathematical fields, such as spectral theory, combinatorics, or discrete complex analysis. In this article, we present some key ideas associated with these processes: we will discuss their definition, the link with Brownian motion in cones, as well as some recent research topics such as the construction of discrete harmonic functions.

math.PR↗

arXiv2022

Stationary Brownian motion in a 3/4-plane: Reduction to a Riemann-Hilbert problem via Fourier transforms

The stationary reflected Brownian motion in a three-quarter plane has been rarely analyzed in the probabilistic literature, in comparison with the quarter plane analogue model. In this context, our main result is to prove that the stationary distribution can indeed be found by solving a boundary value problem of the same kind as the one encountered in the quarter plane, up to various dualities and symmetries. The main idea is to start from Fourier (and not Laplace) transforms, allowing to get a functional equation for a single function of two complex variables.

math.PR↗

arXiv2022

Reflected random walks and unstable Martin boundary

We introduce a family of two-dimensional reflected random walks in the positive quadrant and study their Martin boundary. While the minimal boundary is systematically equal to a union of two points, the full Martin boundary exhibits an instability phenomenon, in the following sense: if some parameter associated to the model is rational (resp.\ non-rational), then the Martin boundary is countable, homeomorphic to $\mathbb Z\cup\{\pm\infty\}$ (resp.\ uncountable, homeomorphic to $\mathbb R\cup\{\pm\infty\}$). Such instability phenomena are very rare in the literature. Along the way of proving this result, we obtain several precise estimates for the Green functions of reflected random walks with escape probabilities along the boundary axes and an arbitrarily large number of inhomogeneity domains. Our methods mix probabilistic techniques and an analytic approach for random walks with large jumps in dimension two.

math.PR↗

arXiv2022

Harmonic functions for singular quadrant walks

We consider discrete (time and space) random walks confined to the quarter plane, with jumps only in directions $(i,j)$ with $i+j \geq 0$ and small negative jumps, i.e., $i,j \geq -1$. These walks are called singular, and were recently intensively studied from a combinatorial point of view. In this paper, we show how the compensation approach introduced in the 90ies by Adan, Wessels and Zijm may be applied to compute positive harmonic functions with Dirichlet boundary conditions. In particular, in case the random walks have a drift with positive coordinates, we derive an explicit formula for the escape probability, which is the probability to tend to infinity without reaching the boundary axes. These formulas typically involve famous recurrent sequences, such as the Fibonacci numbers. As a second step, we propose a probabilistic interpretation of the previously constructed harmonic functions and prove that they allow to compute all positive harmonic functions of these singular walks. To that purpose, we derive the asymptotics of the Green functions in all directions of the quarter plane and use Martin boundary theory.

math.PR↗

arXiv2022

A dual skew symmetry for transient reflected Brownian motion in an orthant

We introduce a transient reflected Brownian motion in a multidimensional orthant, which is either absorbed at the apex of the cone or escapes to infinity. We address the question of computing the absorption probability, as a function of the starting point of the process. We provide a necessary and sufficient condition for the absorption probability to admit an exponential product form, namely, that the determinant of the reflection matrix is zero. We call this condition a dual skew symmetry. It recalls the famous skew symmetry introduced by Harrison, which characterizes the exponential stationary distributions in the recurrent case. The duality comes from that the partial differential equation satisfied by the absorption probability is dual to the one associated with the stationary distribution in the recurrent case.

math.PR↗

arXiv2022

Persistence for a class of order-one autoregressive processes and Mallows-Riordan polynomials

We establish exact formulae for the persistence probabilities of an AR(1) sequence with symmetric uniform innovations in terms of certain families of polynomials, most notably a family introduced by Mallows and Riordan as enumerators of finite labeled trees when ordered by inversions. The connection of these polynomials with the volumes of certain polytopes is also discussed. Two further results provide factorizations of general AR(1) models, one for negative drifts with continuous innovations, and one for positive drifts with continuous and symmetric innovations. The second factorization extends a classical universal formula of Sparre Andersen for symmetric random walks. Our results also lead to explicit asymptotic estimates for the persistence probabilities.

math.PR↗

arXiv2022

Asymptotics of arithmetic functions of GCD and LCM of random integers in hyperbolic regions

We prove limit theorems for the greatest common divisor and the least common multiple of random integers. While the case of integers uniformly distributed on a hypercube with growing size is classical, we look at the uniform distribution on sublevel sets of multivariate symmetric polynomials, which we call hyperbolic regions. Along the way of deriving our main results, we obtain some asymptotic estimates for the number of integer points in these hyperbolic domains, when their size goes to infinity.

math.NT↗

arXiv2021

Lattice walks confined to an octant in dimension 3: (non-)rationality of the second critical exponent

In the field of enumeration of walks in cones, it is known how to compute asymptotically the number of excursions (finite paths in the cone with fixed length, starting and ending points, using jumps from a given step set). As it turns out, the associated critical exponent is related to the eigenvalues of a certain Dirichlet problem on a spherical domain. An important underlying question is to decide whether this asymptotic exponent is a (non-)rational number, as this has important consequences on the algebraic nature of the associated generating function. In this paper, we ask whether such an excursion sequence might admit an asymptotic expansion with a first rational exponent and a second non-rational exponent. While the current state of the art does not give any access to such many-term expansions, we look at the associated continuous problem, involving Brownian motion in cones. Our main result is to prove that in dimension three, there exists a cone such that the heat kernel (the continuous analogue of the excursion sequence) has the desired rational/non-rational asymptotic property. Our techniques come from spectral theory and perturbation theory. More specifically, our main tool is a new Hadamard formula, which has an independent interest and allows us to compute the derivative of eigenvalues of spherical triangles along infinitesimal variations of the angles.

math.CO↗

arXiv2021

The generating function of the survival probabilities in a cone is not rational

A. We look at multidimensional random walks (Sn) n 0 in convex cones, and address the question of whether two naturally associated generating functions may define rational functions. The first series is the one of the survival probabilities P($τ$ > n), where $τ$ is the first exit time from a given cone; the second series is that of the excursion probabilities P($τ$ > n, Sn = y). Our motivation to consider this question is twofold: first, it goes along with a global effort of the combinatorial community to classify the algebraic nature of the series counting random walks in cones; second, rationality questions of the generating functions are strongly associated with the asymptotic behaviors of the above probabilities, which have their own interest. Using wellknown relations between rationality of a series and possible asymptotics of its coefficients, recent probabilistic estimates immediately imply that the excursion generating function is not rational. Regarding the survival probabilities generating function, we propose a short, elementary and selfcontained proof that it cannot be rational neither.

math.CO↗

arXiv2021

Counting quadrant walks via Tutte's invariant method

In the 1970s, William 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 20 years 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, taken in $\{-1, 0,1\}^2$. We first adapt Tutte's approach to prove (or reprove) the algebraicity of all quadrant models known or conjectured to be algebraic. 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 follows almost automatically. Then, we move to a complex analytic viewpoint that has already proved very powerful, leading in particular to integral expressions for 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 for the generating function, and a proof that this series is D-algebraic (that is, satisfies polynomial differential equations).

math.CO↗

arXiv2021

Plane bipolar orientations and quadrant walks

Bipolar orientations of planar maps have recently attracted some interest in combinatorics, probability theory and theoretical physics. Plane bipolar orientations with $n$ edges are known to be counted by the $n$th Baxter number $b(n)$, which can be defined by a linear recurrence relation with polynomial coefficients. Equivalently, the associated generating function $\sum_n b(n)t^n$ is D-finite. In this paper, we address a much refined enumeration problem, where we record for every $r$ the number of faces of degree $r$. When these degrees are bounded, we show that the associated generating function is given as the constant term of a multivariate rational series, and thus is still D-finite. We also provide detailed asymptotic estimates for the corresponding numbers. The methods used earlier to count all plane bipolar orientations, regardless of their face degrees, do not generalize easily to record face degrees. Instead, we start from a recent bijection, due to Kenyon et al., that sends bipolar orientations onto certain lattice walks confined to the first quadrant. Due to this bijection, the study of bipolar orientations meets the study of walks confined to a cone, which has been extremely active in the past 15 years. Some of our proofs rely on recent developments in this field, while others are purely bijective. Our asymptotic results also involve probabilistic arguments.

math.CO↗

arXiv2021

Martin boundary of killed random walks on isoradial graphs

We consider killed planar random walks on isoradial graphs. Contrary to the lattice case, isoradial graphs are not translation invariant, do not admit any group structure and are spatially non-homogeneous. Despite these crucial differences, we compute the asymptotics of the Martin kernel, deduce the Martin boundary and show that it is minimal. Similar results on the grid $\mathbb Z^d$ are derived in a celebrated work of Ney and Spitzer.

math.PR↗

arXiv2020

The $Z$-invariant Ising model via dimers

The $Z$-invariant Ising model (Baxter in Philos Trans R Soc Lond A Math Phys Eng Sci 289(1359):315--346, 1978) is defined on an isoradial graph and has coupling constants depending on an elliptic parameter $k$. When $k=0$ the model is critical, and as $k$ varies the whole range of temperatures is covered. In this paper we study the corresponding dimer model on the Fisher graph, thus extending our papers (Boutillier and de Tili{è}re in Probab Theory Relat Fields 147:379--413, 2010; Commun Math Phys 301(2):473--516, 2011) to the full $Z$-invariant case. One of our main results is an explicit, local formula for the inverse of the Kasteleyn operator. Its most remarkable feature is that it is an elliptic generalization of Boutillier and de Tili{è}re (2011): it involves a local function and the massive discrete exponential function introduced in Boutillier et al. (Invent Math 208(1):109--189, 2017). This shows in particular that $Z$-invariance, and not criticality, is at the heart of obtaining local expressions. We then compute asymptotics and deduce an explicit, local expression for a natural Gibbs measure. We prove a local formula for the Ising model free energy. We also prove that this free energy is equal, up to constants, to that of the $Z$-invariant spanning forests of Boutillier et al. (2017), and deduce that the two models have the same second order phase transition in $k$. Next, we prove a self-duality relation for this model, extending a result of Baxter to all isoradial graphs. In the last part we prove explicit, local expressions for the dimer model on a bipartite graph corresponding to the XOR version of this $Z$-invariant Ising model.

math.PR↗

arXiv2020

Integral expression for the stationary distribution of reflected brownian motion in a wedge

For Brownian motion in a (two-dimensional) wedge with negative drift and oblique reflection on the axes, we derive an explicit formula for the Laplace transform of its stationary distribution (when it exists), in terms of Cauchy integrals and generalized Chebyshev polyno-mials. To that purpose we solve a Carleman-type boundary value problem on a hyperbola, satisfied by the Laplace transforms of the boundary stationary distribution.

math.PR↗