Diffusive geodesics wandering in networks of rigid chains

TL;DR

The paper develops a chain-based spatial network model where successive bond orientations exhibit memory, producing correlated edge geometries. Through simulations of geodesics in a dense regime, it shows a new Euclidean first-passage percolation universality class with wandering exponent $\xi=1/2$ and travel-time fluctuation exponent $\chi=0$, consistent with KPZ but violating Poissonian bounds. Transverse fluctuations align with the Kolmogorov distribution, while travel-time fluctuations are non-Gaussian and non-Tracy–Widom, indicating novel statistical behavior due to orientational correlations. The work introduces the wiggliness observable to quantify geodesic curvature and suggests a persistence-length scale controlling the transition between correlated and Poissonian regimes.

Abstract

We introduce an ensemble of spatial networks built from the junctions of hindered-rotation chains, incorporating directional correlations between bonds, an aspect ignored in the standard network modeling paradigm. The emergent random networks support geodesics with a wandering exponent $ξ= 1/2$, and a travel-time fluctuation exponent $χ= 0$, consistent with the KPZ relation, yet violating the bound~$χ\geq1/8$ predicted in the Poissonian framework. Transverse deviations follow the Kolmogorov distribution, indicating similarities between Brownian bridge excursions and geodesics in a random medium with correlated edges orientations. These results reveal a new universality class of Euclidean first-passage percolation, where local orientational memory reshapes transport properties and challenges existing bounds for random spatial networks.

Figures (17)

• Figure 1: Realizations of networks with closed boundaries for various values of $\theta$ and $L$, at approximately equal density. The coupling between chain length and torsion is expected to affect macroscopic properties, such as the homogeneity of the chain length density. Configurations (a) and (c) show stronger heterogeneity in the domain coverage, while (b) and (d) cover it uniformly.
• Figure 2: Network layout for $L=50$, ${\color{black} \Delta}=50$, open boundaries and various values of $\theta$. Node size is proportional to their betweenness centrality (defined in Appendix \ref{['sec:navigability']}). As the torsion parameter increases, branches are less and less rigid, leading to the formation of peripheral loops.
• Figure 3: Realizations of the model for $\theta=0$ (A), $\theta=0.09$ (B) and $\theta=0.74$ (C), for line density $\rho=3$. The red lines represent geodesics from points close to $(0,h/2)$ and $(W,h/2)$. Notice in (A) and (B) the long 'straight' subsequences of the geodesics, corresponding to navigation on a single chain. Since the persistence length (see Appendix \ref{['sec:scaling']}) of these chains is very high--it grows in $\theta^{-2}$--navigation along a single chain occurs almost rectilinearly.
• Figure 4: Scaling of the average transversal deviation $\mathrm{E}(D)$ (triangles) and of $V(T)^{1/2}$ (squares) in function of the Euclidean distance for $\theta=0$ (left), $\theta=0.09$ (center) and $\theta=0.74$ (right). Each point represents an average over $2000$ realizations, the dashed line is a power-law of exponent $1/2$, and the pointed line is a constant, plotted as a guide for the eye.
• Figure 5: Distribution of normalized fluctuations $\delta$ of the transverse width $D$, for the three torsion parameters considered. The red line corresponds to the rescaled distribution of the maximum of Brownian bridge excursions, defined in \ref{['eq:bb']}.