[Paper Review] On Limit Constants in Last Passage Percolation in Transitive Tournaments

Kunal Dutta|arXiv (Cornell University)|May 20, 2020

Random Matrices and Applications15 references4 citations

TL;DR

This paper resolves the exact value of the limit constant βₜᵣ(p) for last passage percolation on transitive tournaments with i.i.d. Bernoulli(p) edge weights. Using a recurrence relation and bivariate generating functions, it derives βₜᵣ(p) = 1 / Σₙ≥₁ (1−p)^(ⁿ²), proving βₜᵣ(0.5) ≈ 0.60915, and establishes the scaling constants for the functional central limit theorem via singularity analysis and variance asymptotics.

ABSTRACT

We investigate the \emph{last passage percolation} problem on transitive tournaments, in the case when the edge weights are independent Bernoulli random variables. Given a transitive tournament on $n$ nodes with random weights on its edges, the last passage percolation problem seeks to find the weight $X_n$ of the heaviest path, where the weight of a path is the sum of the weights on its edges. We give a recurrence relation and use it to obtain a (bivariate) generating function for the probability generating function of $X_n$. This also gives exact combinatorial expressions for $\mathbb{E}[X_n]$, which was stated as an open problem by Yuster [\emph{Disc. Appl. Math.}, 2017]. We further determine scaling constants in the limit laws for $X_n$. Define $β_{tr}(p) := \lim_{n o \infty} \frac{\mathbb{E}[X_n]}{n-1}$. Using singularity analysis, we show \[ β_{tr}(p) = \left(\sum_{n\geq 1}(1-p)^{n\choose 2} ight)^{-1}. \] In particular, $β_{tr}(0.5) = \left(\sum_{n\geq 1} 2^{-{n\choose 2}} ight)^{-1} = 0.60914971106...$. This settles the question of determining the value of $β_{tr}(0.5)$, initiated by Yuster. $β_{tr}(p)$ is also the limiting value in the strong law of large numbers for $X_n$, given by Foss, Martin, and Schmidt [\emph{Ann. Appl. Probab.}, 2014]. We also derive the scaling constants in the functional central limit theorem for $X_n$ proved by Foss et al.

Motivation & Objective

• To determine the exact value of the limit constant βₜᵣ(p) = limₙ→∞ ℰ[Xₙ]/(n−1) for last passage percolation on transitive tournaments with i.i.d. Bernoulli(p) edge weights.
• To resolve an open problem posed by Yuster (2017) on the exact expectation ℰ[Xₙ] for small n and general n.
• To derive the scaling constants in the functional central limit theorem for Xₙ, confirming and extending results by Foss, Martin, and Schmidt (2014).
• To provide a combinatorial and analytic framework using generating functions and singularity analysis for Bernoulli-distributed edge weights.

Proposed method

• Derive a recurrence relation for the probability generating function of Xₙ, the heaviest path weight in a transitive tournament with n nodes.
• Construct a bivariate generating function Z(x,t) encoding the distribution of Xₙ, enabling exact computation of ℰ[Xₙ] via differentiation.
• Apply singularity analysis to the generating function Bₚ(1) = Σₙ≥₁ (1−p)^(ⁿ²) to extract asymptotic behavior and derive βₜᵣ(p) = 1 / Bₚ(1).
• Compute the variance of Xₙ asymptotically using higher-order coefficients in the generating function expansion.
• Establish the scaling constant for the functional central limit theorem by analyzing the limit of var(Xₙ)/(n−1), yielding σ² = Bₚ(1)⁻²(1 + 6Bₚ′(1)/Bₚ(1) − Bₚ(1)).
• Use pointwise convergence and independence of increments to prove weak convergence to Brownian motion, leveraging the regenerative structure of the process.

Experimental results

Research questions

• RQ1 What is the exact value of the limit constant βₜᵣ(0.5) for last passage percolation on transitive tournaments with i.i.d. Bernoulli(0.5) edge weights?
• RQ2 Can a closed-form expression be derived for ℰ[Xₙ] in transitive tournaments with Bernoulli(p) edge weights?
• RQ3 What are the precise scaling constants in the functional central limit theorem for Xₙ as n → ∞?
• RQ4 How does the generating function approach simplify or extend the regenerative structure analysis of Foss et al. (2014)?
• RQ5 Can the combinatorial and analytic techniques used for Bernoulli weights be generalized to other edge weight distributions, such as uniform or discrete k-valued distributions?

Key findings

• The exact value of βₜᵣ(p) is derived as βₜᵣ(p) = 1 / Σₙ≥₁ (1−p)^(ⁿ²), resolving an open problem posed by Yuster.
• For p = 0.5, βₜᵣ(0.5) = 1 / Σₙ≥₁ 2^(-ⁿ²) ≈ 0.60914971106, providing the exact limit constant for the expected heaviest path weight.
• The strong law of large numbers for Xₙ is confirmed with limₙ→∞ ℰ[Xₙ]/(n−1) = βₜᵣ(p), matching the result of Foss, Martin, and Schmidt (2014).
• The scaling constant for the functional central limit theorem is derived as limₙ→∞ var(Xₙ)/(n−1) = Bₚ(1)⁻²(1 + 6Bₚ′(1)/Bₚ(1) − Bₚ(1)), where Bₚ(1) = Σₙ≥₁ (1−p)^(ⁿ²).
• The paper provides a direct combinatorial proof of the functional limit theorem for Bernoulli weights, avoiding Donsker’s theorem by using singularity analysis and increment independence.
• The method reveals that path weight differences across intervals converge to independent normal variables after proper scaling, supporting the convergence to Brownian motion.