[Paper Review] The speed of frogs with drift on $\mathbb{Z}$

Thomas Höfelsauer, Felizitas Weidner|arXiv (Cornell University)|May 19, 2015

Stochastic processes and statistical mechanics6 references3 citations

TL;DR

This paper studies the frog model with drift on the integer lattice ℤ, proving that the speed of the minimum position of active frogs equals 2p−1 for p>1/2, while the speed of the maximum is strictly increasing and less than 1 for p<1. It further establishes a limit theorem showing the empirical distribution of active frog positions, when rescaled, converges weakly to the uniform distribution on [0,1].

ABSTRACT

In this article we consider the frog model with drift on $\mathbb{Z}$ and investigate the behaviour of the cloud of the frogs. In particular, we show that the speed of the minimum equals the speed of a single frog and prove some properties of the speed of the maximum. In addition, we show a limit theorem for the empirical distribution.

Motivation & Objective

• To analyze the asymptotic speed of the leftmost (minimum) and rightmost (maximum) active frogs in the frog model with drift on ℤ.
• To investigate the distribution of active frogs relative to the frontiers of the occupied set over time.
• To establish a limit theorem for the empirical distribution of active frog positions under appropriate scaling.
• To explore the functional dependence of the maximum speed on the drift parameter p, particularly its monotonicity and potential concavity.

Proposed method

• Uses Liggett’s Subadditive Ergodic Theorem to prove almost sure existence of the speed of the maximum and the activation time limit.
• Applies coupling arguments and concentration inequalities to control fluctuations in frog positions and activation times.
• Defines rescaled empirical measures μₙ to study the spatial distribution of active frogs relative to the minimum and maximum positions.
• Employs large deviation estimates and asymptotic analysis of activation times Tᵢ to bound the range of active frog positions.
• Uses the convergence of Tᵢ/i to vₘₐₓ⁻¹ to derive bounds on the rescaled positions of frogs initially at site i.
• Applies weak convergence arguments to show that the rescaled empirical measure μₙ converges almost surely to Lebesgue measure on [0,1].

Experimental results

Research questions

• RQ1 What is the exact value of the speed of the minimum position of active frogs in the frog model with drift p>1/2?
• RQ2 How does the speed of the maximum position depend on the drift parameter p?
• RQ3 Does the empirical distribution of active frog positions, when rescaled between the minimum and maximum, converge to a uniform distribution?
• RQ4 Is the speed of the maximum a concave function of p, as suggested by simulations?

Key findings

• The speed of the minimum, vₘᵢₙ, is exactly 2p−1 for all p>1/2.
• The speed of the maximum, vₘₐₓ, is strictly increasing in p and satisfies vₘₐₓ<1 for all p<1.
• The rescaled empirical distribution μₙ of active frog positions converges weakly almost surely to the Lebesgue measure on [0,1] as n→∞.
• The number of active frogs grows linearly with time, with limₙ→∞ n/|Aₙ| = vₘₐₓ⁻¹ for p>1/2.
• The minimum and maximum positions grow linearly almost surely, with limₙ→∞ Mₙ/n = vₘₐₓ and limₙ→∞ mₙ/n = vₘᵢₙ.
• The paper conjectures that vₘₐₓ is a concave function of p, based on simulations and heuristic reasoning.