Cavity and replica methods for the spectral density of sparse symmetric random matrices

We review the problem of how to compute the spectral density of sparse symmetric random matrices, i.e. weighted adjacency matrices of undirected graphs. Starting from the Edwards-Jones formula, we illustrate the milestones of this line of research, including the pioneering work of Bray and Rodgers using replicas. We focus first on the cavity method, showing that it quickly provides the correct recursion equations both for single instances and at the ensemble level. We also describe an alternative replica solution that proves to be equivalent to the cavity method. Both the cavity and the replica derivations allow us to obtain the spectral density via the solution of an integral equation for an auxiliary probability density function. We show that this equation can be solved using a stochastic population dynamics algorithm, and we provide its implementation. In this formalism, the spectral density is naturally written in terms of a superposition of local contributions from nodes of given degree, whose role is thoroughly elucidated. This paper does not contain original material, but rather gives a pedagogical overview of the topic. It is indeed addressed to students and researchers who consider entering the field. Both the theoretical tools and the numerical algorithms are discussed in detail, highlighting conceptual subtleties and practical aspects.

Counting individual bacteria during high-throughput growth in microfluidic droplets

Bacterial growth in microfluidic droplets is relevant in biotechnology, in microbial ecology, and in understanding stochastic population dynamics in small populations. However, it has proved challenging to automate measurement of absolute bacterial numbers within droplets, forcing the use of proxy measures for population size. Here we present a microfluidic device and imaging protocol that allows high-resolution imaging of thousands of droplets, such that individual bacteria stay in the focal plane and can be counted automatically. We demonstrate the use of this approach to track stochastic growth of multiple replicate Escherichia coli populations within droplets.