Limit Theorems for the Maximal Path Weight in a Directed Graph on the Line with Random Weights of Edges

Konstantopoulos, Takis; Logachov, A. V.; Mogulskii, A. A.; Foss, Sergey

doi:10.1134/s0032946021020058

Cited by 1 publication

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“…Concerning other types of behavior of maximal lengths or weights, e.g., in the large deviations sense, we mention that, when weights have a distribution supported on the integers, normal and moderate large deviations were obtained in [65]. In addition local limit theorems were obtained when the weight distribution is non-lattice.…”

Section: Analytic Properties Of Charged Graphsmentioning

confidence: 94%

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Estimation of the last passage percolation constant in a charged complete directed acyclic graph via perfect simulation

Foss

Konstantopoulos²,

Mallein

et al. 2023

ALEA

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Our object of study is the asymptotic growth of heaviest paths in a charged (weighted with signed weights) complete directed acyclic graph. Edge charges are i.i.d. random variables with common distribution F supported on [−∞, 1] with essential supremum equal to 1 (a charge of −∞ is understood as the absence of an edge). The asymptotic growth rate is a constant that we denote by C(F ). Even in the simplest case where F = pδ 1 + (1 − p)δ −∞ , corresponding to the longest path in the Barak-Erdős random graph, there is no closed-form expression for this function, but good bounds do exist. In this paper we construct a Markovian particle system that we call "Max Growth System" (MGS), and show how it is related to the charged random graph. The MGS is a generalization of the Infinite Bin Model that has been the object of study of a number of papers. We then identify a random functional of the process that admits a stationary version and whose expectation equals the unknown constant C(F ). Furthermore, we construct an effective perfect simulation algorithm for this functional which produces samples from the random functional.

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Section: Analytic Properties Of Charged Graphsmentioning

confidence: 94%

“…To do this we discuss an auxiliary Markovian particle system, called Max Growth System (MGS), that is an analog of the Infinite Bin Model. Our reference throughout this section is [65].…”

Section: Perfect Simulation Aspects Of Charged Graphsmentioning

confidence: 99%