Emergent correlations in the selected link-times along optimal paths

TL;DR

This work analyzes the statistics of selected link-times (SLTs) along optimal paths in weakly disordered first-passage percolation on a square lattice. By combining KPZ scaling, Tracy-Widom fluctuations, and a non-Gaussian, non-Wick correlation structure, it shows that conditioning on geodesics induces long-range correlations among LT samples, driving the SLT sum to follow a Tracy-Widom distribution rather than Gaussian behavior. The authors identify universal power-law decays in global and local SLT moments, demonstrate directional effects between axial and diagonal paths, and propose a conformal-field-theory–inspired framework to account for higher-order correlations. These findings link emergent correlations in random media to KPZ universality and full-counting statistics, with implications for understanding transport in disordered systems and the limits of the central limit theorem in conditioned ensembles.

Abstract

In the context of first-passage percolation (FPP), we investigate the statistical properties of the selected link-times (SLTs) -the random link times comprising the optimal paths (or geodesics) connecting two given points. We focus on weakly disordered square lattices, whose geodesics are known to fall under the Kardar-Parisi-Zhang (KPZ) universality class. Our analysis reveals universal power-law decays with the end-to-end distance for both the average and standard deviation of the SLTs, along with an intricate pattern of long-range correlations, whose scaling exponents are directly linked to KPZ universality. Crucially, the SLT distributions for diagonal and axial paths exhibit significant differences, which we trace back to the distinct directed and undirected nature, respectively, of the underlying geodesics. Moreover, we demonstrate that the SLT distribution violates the conditions of the central limit theorem. Instead, SLT sums follow the Tracy-Widom distribution characteristic of the KPZ class, which we associate with evidence for the emergence of high-order long-range correlations in the ensemble.

Figures (11)

• Figure 1: (a) Asymptotic value of the global SLT mean as as function of the CV for the two distributions and the two lattice directions. The broken line indicates the LT mean. (b) Convergence of the SLT averages towards their asymptotic values for geodesics along the diagonal, as we increase the end-to-end distance $d$. (c) Same data, along the axis. The broken line in both panels represents power-law behavior with exponent $-1/z$.
• Figure 2: (a) Asymptotic value of the global SLT deviation as function of the CV for the two distributions and the two lattice directions. The broken line indicates the LT deviation. (b) Convergence of the SLT deviations towards their asymptotic values for geodesics along the diagonal, as we increase the distance $d$. (c) Same data, along the axis. The broken line in both panels represents power-law behavior with exponent $-1/z$.
• Figure 3: Histograms of $\chi$, defined in Eq. \ref{['eq:fig_chi']}, which is a standardized version of the arrival time $T$, for the maximum spanning distance $d_\text{max}$ along (a) the diagonal and (b) the axis, and different LT distributions. The suitably rescaled and reversed TW-GUE distribution is shown for comparison.
• Figure 4: (a)-(h) SLT histograms, $\hat{f}(t)$, for geodesics reaching the maximum distance $d_\text{max}$ along the axis and the diagonal, using different LT distributions, along with the original LT probability density $f(t)$. Left and right columns show uniform and Weibull cases respectively, with CV growing from top to bottom. (i) Standardized SLT histograms for the uniform distribution along the axis and the diagonal (inset) directions, the horizontal line representing the original LT distribution. For the sake of comparison, we have displayed in the main figure (axis), the curve of the inset (diagonal) corresponding to CV=$0.57$.
• Figure 5: (a) Schematic representation of the probability shift between the SLT distribution (solid line) and the original LT distribution (horizontal red line). (b) KS distance between the LT and the SLT distributions for asymptotically large distances along the axis and the diagonal, as a function of the CV for the two LT distributions. (c) Convergence of the KS distances between the LT and the SLT distributions toward their asymptotic values as a function of the distance for the diagonal direction. (d) Same data, for the axis. The broken line in both panels represents power-law behavior with exponent $-1/z$.