[Paper Review] Lower large deviations and laws of large numbers for maximal flows through a box in first passage percolation

Raphaël Rossignol, Marie Théret|ArXiv.org|Jan 7, 2008

Stochastic processes and statistical mechanics3 citations

TL;DR

This paper establishes lower large deviation principles for maximal flow in first passage percolation on $\mathbb{Z}^d$, showing exponential decay of probabilities that rescaled flows $\tau/s$ and $\phi/s$ are abnormally small, with speed proportional to surface area $s$. It proves laws of large numbers for both flows under optimal moment conditions, using almost subadditivity and concentration inequalities, extending prior results to general box orientations and weaker moment assumptions.

ABSTRACT

We consider the standard first passage percolation model in $\mathbb{Z}^d$ for $d\geq 2$. We are interested in two quantities, the maximal flow $τ$ between the lower half and the upper half of the box, and the maximal flow $ϕ$ between the top and the bottom of the box. A standard subadditive argument yields the law of large numbers for $τ$ in rational directions. Kesten and Zhang have proved the law of large numbers for $τ$ and $ϕ$ when the sides of the box are parallel to the coordinate hyperplanes: the two variables grow linearly with the surface $s$ of the basis of the box, with the same deterministic speed. We study the probabilities that the rescaled variables $τ/s$ and $ϕ/s$ are abnormally small. For $τ$, the box can have any orientation, whereas for $ϕ$, we require either that the box is sufficiently flat, or that its sides are parallel to the coordinate hyperplanes. We show that these probabilities decay exponentially fast with $s$, when $s$ grows to infinity. Moreover, we prove an associated large deviation principle of speed $s$ for $τ/s$ and $ϕ/s$, and we improve the conditions required to obtain the law of large numbers for these variables.

Motivation & Objective

• To establish lower large deviation principles for maximal flow $\tau$ and $\phi$ in first passage percolation on $\mathbb{Z}^d$ for general box orientations and height functions.
• To improve the moment conditions required for the law of large numbers for $\tau$ and $\phi$, particularly for irrational directions and non-straight boxes.
• To prove that lower large deviation probabilities decay exponentially fast with surface area $s$, establishing a large deviation principle at speed $s$.
• To extend Kesten and Zhang's results on the law of large numbers for $\phi$ to more general box geometries and weaker moment assumptions.
• To show that the rate function for $\phi$ matches that of $\tau$ under suitable conditions, leveraging existing upper deviation results.

Proposed method

• Uses almost subadditivity of the maximal flow $\tau_n$ to circumvent the loss of subadditivity in irrational directions.
• Applies concentration inequalities and exponential moment bounds to control lower tail probabilities of rescaled flows.
• Employs a reduction to bounded capacity distributions via truncation, followed by coupling arguments to extend results to general distributions.
• Relies on a key proposition (Proposition 6.1) showing that $\phi$ and $\tau$ share the same rate function under mild conditions.
• Uses the large deviation principle for upper tails of $\phi$ from Théret (2007) to complete the lower tail analysis via symmetry and duality.
• Applies Fatou’s lemma and uniform integrability to upgrade almost sure convergence to $L^1$ convergence for the rescaled flows.

Experimental results

Research questions

• RQ1 What is the decay rate of the probability that the rescaled maximal flow $\tau/s$ is significantly smaller than its asymptotic value, for general box orientations?
• RQ2 Under what moment conditions does the law of large numbers hold for $\tau_n$ in irrational directions, where subadditivity fails?
• RQ3 Can the lower large deviation principle for $\phi_n$ be established for non-straight, flat cylinders or general hyperrectangular boxes?
• RQ4 Is the rate function for lower deviations of $\phi_n$ identical to that of $\tau_n$ under the same conditions?
• RQ5 Can the moment condition required for the law of large numbers of $\phi_n$ be weakened compared to Kesten and Zhang’s original result?

Key findings

• The probability that $\tau_n/s$ is abnormally small decays exponentially fast with surface area $s$, for any box orientation.
• For $\phi_n$, lower large deviation probabilities decay exponentially fast when the box is flat or has sides parallel to coordinate hyperplanes.
• A large deviation principle of speed $s$ is established for both $\tau_n/s$ and $\phi_n/s$, with the same rate function under appropriate conditions.
• The law of large numbers for $\tau_n$ holds in any fixed direction (including irrational) under the optimal moment condition $\mathbb{E}[t(e)^{d-1}] < \inyfty$.
• For $\phi_n$, the law of large numbers holds under a weaker moment condition and a height condition $\log h(n)/n^{d-1} \to 0$, improving upon Kesten and Zhang’s result.
• The rate function for $\phi_n$ is shown to coincide with that of $\tau_n$ under the same conditions, confirming a deep connection between the two flows.