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

Abstract The Pitman–Yor process is a random discrete measure. The random weights or masses follow the two-parameter Poisson–Dirichlet distribution with parameters 0 < α < 1, θ > -α. The parameters α and θ correspond to the stable and gamma components, respectively. The distribution of atoms is given by a probability η. In this paper we consider the limit theorems for the Pitman–Yor process and the two-parameter Poisson–Dirichlet distribution. These include the law of large numbers, fluctuations, and moderate or large deviation principles. The limiting procedures involve either α tending to 0 or 1. They arise naturally in genetics and physics such as the asymptotic coalescence time for explosive branching process and the approximation to the generalized random energy model for disordered systems.

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Convergence of directed random graphs to the Poisson-weighted infinite tree

Journal of Applied Probability ◽

10.1017/jpr.2016.13 ◽

2016 ◽

Vol 53 (2) ◽

pp. 463-474 ◽

Cited By ~ 1

Author(s):

Katja Gabrysch

Keyword(s):

Random Graph ◽

Directed Graph ◽

Random Graphs ◽

Limit Theorems ◽

Directed Edge ◽

Weighted Distance ◽

Infinite Tree ◽

Longest Path

Abstract We consider a directed graph on the integers with a directed edge from vertex i to j present with probability n-1, whenever i < j, independently of all other edges. Moreover, to each edge (i, j) we assign weight n-1(j - i). We show that the closure of vertex 0 in such a weighted random graph converges in distribution to the Poisson-weighted infinite tree as n → ∞. In addition, we derive limit theorems for the length of the longest path in the subgraph of the Poisson-weighted infinite tree which has all vertices at weighted distance of at most ρ from the root.

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An Analysis of the Height of Tries with Random Weights on the Edges

Combinatorics Probability Computing ◽

10.1017/s0963548307008796 ◽

2008 ◽

Vol 17 (2) ◽

pp. 161-202 ◽

Cited By ~ 1

Author(s):

N. BROUTIN ◽

L. DEVROYE

Keyword(s):

Search Time ◽

Finite Alphabet ◽

Maximal Path ◽

Processing Times ◽

Random Weights

We analyse the weighted height of random tries built from independent strings of i.i.d. symbols on the finite alphabet {1, . . .d}. The edges receive random weights whose distribution depends upon the number of strings that visit that edge. Such a model covers the hybrid tries of de la Briandais and the TST of Bentley and Sedgewick, where the search time for a string can be decomposed as a sum of processing times for each symbol in the string. Our weighted trie model also permits one to study maximal path imbalance. In all cases, the weighted height is shown to be asymptotic toclognin probability, wherecis determined by the behaviour of thecoreof the trie (the part where all nodes have a full set of children) and the fringe of the trie (the part of the trie where nodes have only one child and formspaghetti-like trees). It can be found by maximizing a function that is related to the Cramér exponent of the distribution of the edge weights.

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