Statistical Mechanics of Complex, Glassy and Non-Equilibrium Systems

Spectral partitioning in random regular blockmodels Paolo Barucca (Zurich/LIMS) Graph partitioning problems emerge in a wide variety of complex systems, ranging from biology to finance, but can be rigorously analysed and solved only for a few graph ensembles. In this work, an ensemble of random graphs with regular block structure, also named equitable graphs, is discussed and analytically studied. In particular, the spectral density of such graphs is computed exactly for a modular, bipartite and core-periphery structure. McKay’s law for random regular graphs is found analytically to apply also for regular modular and bipartite structures when blocks are homogeneous. In core-periphery structures, where blocks are intrinsically heterogeneous, a new law is found to apply for the spectral density. Exact solution to the problem of inferring the original block structure is provided for the models discussed. All analytical results show perfect agreement with numerical experiments. In the framework of equitable graphs the picture of the detectability threshold for naive spectral clustering, i.e. only using the eigenvector associated to the second largest eigenvalue of the adjacency matrix, emerges distinctly, as well as the crucial role of eigenvectorss statistics. Insights from equitable graphs could be used to develop new spectral methods based on both eigenvalues and eigenvectors properties. Shape universality classes in the random sequential adsorption of non-spherical particles Adrian Baule (QMUL) Abstract: Random sequential adsorption (RSA) of particles of a particular shape is used in a large variety of contexts to model particle aggregation and jamming. A key feature of these models is the observed algebraic time dependence of the asymptotic jamming coverage ∼ t−ν as t→∞. However, the exact value of the critical exponent ν is not known apart from the simplest case of the RSA of monodisperse spheres adsorbed on a line (Renyi’s seminal ‘car parking problem’), where ν = 1 can be derived analytically. Empirical simulation studies have conjectured on a case-by-case basis that for general non-spherical particles ν = 1/(d+ d̃), where d denotes the dimension of the domain and d̃ the number of orientational degrees of freedom of a particle. Here, we solve this problem analytically for the d = 1 case — the ‘Paris car parking problem [1]. We prove in particular that the scaling exponent depends on particle shape, contrary to the original conjecture, and falls into two universality classes: (i) ν = 1/(1 + d̃/2) for smooth convex shapes with non-zero curvature, e.g., ellipsoids; (ii) ν = 1/(1 + d̃) for shapes with flat sides, e.g., spherocylinders and polyhedra. The exact solution explains in particular why many empirically observed scalings fall in between these two limits. [1] A. Baule, arXiv:1611.03034 (2016) Free fermions and the classical compact groups Fabio Deelan Cunden (University of Bristol) We unveil the existence of a precise mapping between the ground state of non-interacting free fermions in a box with classical (absorbing, reflecting, and periodic) boundary conditions and the eigenvalue statistics of the classical compact groups. The associated determinantal point processes can be extended in two natural directions: i) we consider the full family of admissible quantum