Random Walks in the Quarter-Plane

Fayolle, Guy; Iasnogorodski, Roudolf; Малышев, В. А.

doi:10.1007/978-3-642-60001-2

Cited by 132 publications

(270 citation statements)

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“…Random walks conditioned to live in domains C ⊂ Z d are of growing interest because of the range of their applications in enumerative combinatorics, in probability theory and in harmonic analysis (cf. [7], [9], [11], [17], [18], [30]). Doob h-transforms, where h is harmonic for the random walk, positive within C and vanishing on its boundary ∂C, are used to perform such conditioning.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Discrete harmonic functions in Lipschitz domains

Mustapha¹,

Sifi²

2019

Electron. Commun. Probab.

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We prove the existence and uniqueness of a discrete nonnegative harmonic function for a random walk satisfying finite range, centering and ellipticity conditions, killed when leaving a globally Lipschitz domain in Z d . Our method is based on a systematic use of comparison arguments and discrete potential-theoretical techniques.2010 Mathematics Subject Classification: Primary 60G50, 31C35; Secondary 60G40, 30F10.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Discrete harmonic functions in Lipschitz domains

Mustapha¹,

Sifi²

2019

Electron. Commun. Probab.

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“…In modern times, a large area of work on this topic centers around using the so-called kernel method to express generating functions of large classes of models as positive series extractions or diagonals of multivariate rational functions, proving that the generating functions are (or are not) D-finite; that is, determining whether they satisfy a linear differential equation with polynomial coefficients. Although there are a finite number of models with steps in {±1, 0} 2 that are restricted to a quarter-plane, their combinatorics has been the subject of intense study in recent years [21,11,31,41,13,7,8,50,42,37,38,4,12,2,6,19]. The models considered in this paper are higher-dimensional generalizations of this two-dimensional setting.…”

Section: Lattice Walks In Restricted Regionsmentioning

confidence: 99%

“…In one approach, developed in part for problems arising in queuing theory, a singularity analysis of solutions to functional equations satisfied by lattice path generating functions yields analytic and asymptotic information. The text of Fayolle et al [21] gives a detailed view on the techniques involved, some of which inspired Bousquet-Mélou's creation of the algebraic kernel method; see also Malyšev [35] for an early history. The lattice path models we study have asymptotics of the form C n α ρ n for constants α and ρ, where C is constant or depends only on the periodicity of n. Fayolle and Raschel [22] used these techniques to outline a method that, in principle, allows one to calculate the exponential growth ρ for many quadrant models.…”

Section: Past Workmentioning

confidence: 99%

“…We closely follow the kernel method as outlined in Bousquet-Mélou and Mishna [13], and a straightforward generalization to higher dimensions discussed by Melczer and Mishna [38]. This approach builds heavily on a probabilistic framework detailed by Fayolle, Iasnogorodski, and Malyshev [21]; see also Bousquet-Mélou and Petkovšek [15] for a more general overview on the kernel method.…”

Section: Lattice Path Generating Functionsmentioning

confidence: 99%

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Higher Dimensional Lattice Walks: Connecting Combinatorial and Analytic Behavior

Melczer¹

2019

SIAM J. Discrete Math.

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We consider the enumeration of walks on the non-negative lattice N d , with steps defined by a set S ⊂ {−1, 0, 1} d \{0}. Previous work in this area has established asymptotics for the number of walks in certain families of models by applying the techniques of analytic combinatorics in several variables (ACSV), where one encodes the generating function of a lattice path model as the diagonal of a multivariate rational function. Melczer and Mishna obtained asymptotics when the set of steps S is symmetric over every axis; in this setting one can always apply the methods of ACSV to a multivariate rational function whose set of singularities is a smooth manifold (the simplest case). Here we go further, providing asymptotics for models with generating functions that must be encoded by multivariate rational functions having non-smooth singular sets. In the process, our analysis connects past work to deeper structural results in the theory of analytic combinatorics in several variables. One application is a closed form for asymptotics of models defined by step sets that are symmetric over all but one axis. As a special case, we apply our results when d = 2 to give a rigorous proof of asymptotics conjectured by Bostan and Kauers; asymptotics for walks returning to boundary axes and the origin are also given.

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“…To prove that this chain is ergodic for k > 3, we use Theorem 1.2.1 from [19], henceforth referred to as Malyshev's result, which defines expected jumps 3 by…”

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confidence: 99%

On the Stochastic Analysis of a Quantum Entanglement Switch

Vardoyan

Guha

Nain

et al. 2019

SIGMETRICS Perform. Eval. Rev.

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We study a quantum entanglement switch that serves k users in a star topology. We model variants of the system using Markov chains and standard queueing theory and obtain expressions for switch capacity and the expected number of qubits stored in memory at the switch. While it is more accurate to use a discrete-time Markov chain (DTMC) to model such systems, we quickly encounter practical constraints of using this technique and switch to using continuous-time Markov chains (CTMCs). Using CTMCs allows us to obtain a number of analytic results for systems in which the links are homogeneous or heterogeneous and for switches that have infinite or finite buffer sizes. In addition, we can model the effects of decoherence of quantum states fairly easily using CTMCs. We also compare the results we obtain from the DTMC against the CTMC in the case of homogeneous links and infinite buffer, and learn that the CTMC is a reasonable approximation of the DTMC. From numerical observations, we discover that decoherence has little effect on capacity and expected number of stored qubits for homogeneous systems. For heterogeneous systems, especially those operating close to stability constraints, buffer size and decoherence can have significant effects on performance metrics. We also learn that in general, increasing the buffer size from one to two qubits per link is advantageous to most systems, while increasing the buffer size further yields diminishing returns. Index Terms-entanglement, quantum switch, Markov chain. arXiv:1903.04420v2 [quant-ph] 18 Jul 2019 1 With a linear optical circuit, four unentangled ancilla single photons and photon number resolving detectors, with all the devices being lossless, q = 25/32 = 0.78 can be achieved for BSMs [16]. With other technologies q close to 1 can be achieved [17].

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Random Walks in the Quarter-Plane

Cited by 132 publications

References 0 publications

Discrete harmonic functions in Lipschitz domains

Discrete harmonic functions in Lipschitz domains

Higher Dimensional Lattice Walks: Connecting Combinatorial and Analytic Behavior

On the Stochastic Analysis of a Quantum Entanglement Switch

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