Emergent correlations in the selected link-times along optimal paths

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.

💡 Research Summary

In this work the authors investigate the statistical properties of the selected link‑times (SLTs) that compose optimal paths (geodesics) in the first‑passage percolation (FPP) model on a weakly disordered square lattice. Each edge of the lattice carries a positive crossing time drawn independently from a prescribed distribution; two families of distributions are considered: a uniform distribution on a finite interval and a Weibull distribution, both characterized by a mean τ=5 and a tunable coefficient of variation CV=σ/τ. Simulations are performed on a (2L+1)×(2L+1) lattice with L=1005, using N_s=2×10⁴ independent disorder realizations, and the maximal Euclidean distances examined are d_max=1000 along the lattice axes and d_max=1000√2 along the diagonal.

The paper first characterizes the global SLT distribution ˆf(t). Because geodesics preferentially traverse edges with low crossing times, the mean of the SLTs, ˆτ(d), is always smaller than the original mean τ and decays with the end‑to‑end distance d. The decay follows a power law ˆτ(d)−ˆτ(∞)∝d^{−α_τ} with an exponent α_τ≈2/3, which coincides with the inverse dynamic exponent 1/z of the Kardar‑Parisi‑Zhang (KPZ) universality class (z=3/2). The standard deviation ˆσ(d) exhibits the same scaling, ˆσ(d)−ˆσ(∞)∝d^{−α_σ} with α_σ≈2/3. The asymptotic values ˆτ(∞) and ˆσ(∞) decrease as CV grows; for small CV the shift of ˆτ(∞) from τ is more pronounced along the axis than along the diagonal, reflecting the directed nature of diagonal geodesics versus the undirected nature of axial ones. At large CV the anisotropy disappears.

A central result concerns the sum of SLTs, i.e., the arrival time T between two points separated by distance d. If the SLTs were independent samples from the original link‑time distribution, the Central Limit Theorem (CLT) would predict σ_T∝d^{1/2} and Gaussian fluctuations. Instead, the authors find σ_T∝d^{β} with β=1/3, the KPZ growth exponent, and the standardized variable χ=(T−⟨T⟩)/σ_T follows a Tracy‑Widom distribution of the Gaussian Unitary Ensemble (TW‑GUE), albeit reflected horizontally because χ exhibits negative skewness. This demonstrates that the arrival‑time fluctuations belong to the KPZ universality class and that the CLT is violated.

The authors also explore directional differences. Along the diagonal (directed geodesics) the SLT distribution differs from that along the axis (undirected geodesics). In the axial case the geodesic is non‑degenerate in the homogeneous limit, leading to a lower average SLT for small CV. As disorder increases, the two directions converge, indicating that the underlying geometry dominates over directionality at strong disorder.

To explain the CLT violation, the paper measures two‑point and higher‑order correlation functions among SLTs. Both the two‑point correlator C₂(r)=⟨δt_i δt_{i+r}⟩ and the four‑point correlator display long‑range, non‑Wick scaling consistent with a conformal‑field‑theory‑inspired Ansatz. These emergent correlations are directly linked to the KPZ scaling exponents and provide a mechanism for the observed non‑Gaussian statistics of T.

Finally, the full SLT histograms ˆf(t) are compared with the original link‑time density f(t). The SLT histograms are shifted toward lower values and, especially for Weibull disorder, exhibit a strongly suppressed tail. This reflects a selection bias: optimal paths preferentially pick “fast” edges, thereby reshaping the underlying distribution.

In summary, the study demonstrates that in weakly disordered FPP on a square lattice, the selected link‑times form a non‑trivial statistical ensemble characterized by (i) universal power‑law decay of mean and variance with distance, (ii) KPZ‑class Tracy‑Widom fluctuations of total arrival time, (iii) direction‑dependent SLT distributions, and (iv) emergent long‑range, non‑Wick correlations that invalidate the Central Limit Theorem. The work bridges concepts from optimal‑path theory, KPZ universality, and full‑counting statistics, and suggests future extensions to stronger disorder regimes, other lattice topologies, and experimental realizations in transport, fluid flow, or fracture phenomena.