Wavelength selection beyond Turing

Spatial patterns arising spontaneously due to internal processes are ubiquitous in nature, varying from regular patterns of dryland vegetation to complex structures of bacterial colonies. Many of these patterns can be explained in the context of a Turing instability, where patterns emerge due to two locally interacting components that diffuse with different speeds in the medium. Turing patterns are multistable, such that many different patterns with different wavelengths are possible for the same set of parameters, but in a given region typically only one such wavelength is dominant. In the Turing instability region, random initial conditions will mostly lead to a wavelength that is similar to that of the leading eigenvector that arises from the linear stability analysis, but when venturing beyond, little is known about the pattern that will emerge. Using dryland vegetation as a case study, we use different models of drylands ecosystems to study the wavelength pattern that is selected in various scenarios beyond the Turing instability region, focusing the phenomena of localized states and repeated local disturbances

Card shuffling and diophantine approximation

The ``overlapping-cycles shuffle'' mixes a deck of $n$ cards by moving either the $n$th card or the $(n-k)$th card to the top of the deck, with probability half each. We determine the spectral gap for the location of a single card, which, as a function of $k$ and $n$, has surprising behavior. For example, suppose $k$ is the closest integer to $\alpha n$ for a fixed real $\alpha\in(0,1)$. Then for rational $\alpha$ the spectral gap is $\Theta(n^{-2})$, while for poorly approximable irrational numbers $\alpha$, such as the reciprocal of the golden ratio, the spectral gap is $\Theta(n^{-3/2})$.Comment: Published in at http://dx.doi.org/10.1214/07-AAP484 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Noise Sensitivity of the Minimum Spanning Tree of the Complete Graph

We study the noise sensitivity of the minimum spanning tree (MST) of the $n$-vertex complete graph when edges are assigned independent random weights. It is known that when the graph distance is rescaled by $n^{1/3}$ and vertices are given a uniform measure, the MST converges in distribution in the Gromov-Hausdorff-Prokhorov (GHP) topology. We prove that if the weight of each edge is resampled independently with probability $\varepsilon\gg n^{-1/3}$, then the pair of rescaled minimum spanning trees, before and after the noise, converges in distribution to independent random spaces. Conversely, if $\varepsilon\ll n^{-1/3}$, the GHP distance between the rescaled trees goes to $0$ in probability. This implies the noise sensitivity and stability for every property of the MST seen in the scaling limit, e.g., whether the diameter exceeds its median. The noise threshold of $n^{-1/3}$ coincides with the critical window of the Erd\H{o}s-R\'enyi random graphs. In fact, these results follow from an analog theorem we prove regarding the minimum spanning forest of critical random graphs

The String of Diamonds Is Tight for Rumor Spreading

For a rumor spreading protocol, the spread time is defined as the first time that everyone learns the rumor. We compare the synchronous push&pull rumor spreading protocol with its asynchronous variant, and show that for any n-vertex graph and any starting vertex, the ratio between their expected spread times is bounded by O(n^{1/3} log^{2/3} n). This improves the O(sqrt n) upper bound of Giakkoupis, Nazari, and Woelfel (in Proceedings of ACM Symposium on Principles of Distributed Computing, 2016). Our bound is tight up to a factor of O(log n), as illustrated by the string of diamonds graph