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In the first paper of this series [S. Torquato, J. Chem. Phys. {\bf 136}, 054106 (2012)], analytical results concerning the continuum percolation of overlapping hyperparticles in $d$-dimensional Euclidean space $\mathbb{R}^d$ were obtained,…
We consider site (vertex) percolation on $d$-regular graphs, for both constant-degree and growing-degree cases. We give sufficient, and relatively tight, conditions for the emergence of the ``Erd\H{o}s-R\'enyi component phenomenon" in the…
Let a random geometric graph be defined in the supercritical regime for the existence of a unique infinite connected component in Euclidean space. Consider the first-passage percolation model with independent and identically distributed…
The propagation of waves through transmission eigenchannels in complex media is emerging as a new frontier of condensed matter and wave physics. A crucial step towards constructing a complete theory of eigenchannels is to demonstrate their…
In this work we show that under specific anomalous diffusion conditions, chemical systems can produce well-ordered self-similar concentration patterns through a diffusion-driven instability. We also find spiral patterns and patterns with…
Chaos as typical property of non-linear systems has revealed its crucial role in various problems of astrophysics and cosmology. The problems discussed at these lectures include planetary dynamics, galactic dynamics, reconstruction of the…
Global physical properties of random media change qualitatively at a percolation threshold, where isolated clusters merge to form one infinite connected component. The precise knowledge of percolation thresholds is thus of paramount…
We study the length of cycles in the model of spatial random permutations in Euclidean space. In this model, for given length $L$, density $\rho$, dimension $d$ and jump density $\varphi$, one samples $\rho L^d$ particles in a…
The intrinsic geometry of the critical percolation cluster induced by the level set of the metric Gaussian free field on $\mathbb{Z}^{d}$ has been the subject of much recent activity. (Lupu, 2016) established that the critical percolation…
Chaotic internal degrees of freedom of a molecule can act as noise and affect the diffusion of the molecule on a substrate. A separation of time scales between the fast internal dynamics and the slow motion of the centre of mass on the…
We consider the standard model of i.i.d. bond percolation on $\mathbb Z^d$ of parameter $p$. When $p>p_c$, there exists almost surely a unique infinite cluster $\mathcal C_p$. Using the recent techniques of Cerf and Dembin, we prove that…
We study the superconducting proximity effect in inhomogeneous systems in which a disordered or quasicrystalline normal-state wire is connected to a BCS superconductor. We self-consistently compute the local superconducting order parameters…
We present a method to determine the critical rotational frequencies for superfluidity of both uniform and trapped Fermi gases across wide Feshbach resonance. It is found that as one approaches the resonance from the BCS side, beyond a…
Recently it has been demonstrated that the connectivity transition from microscopic connectivity to macroscopic connectedness, known as percolation, is generically announced by a cascade of microtransitions of the percolation order…
We investigate the connections between microscopic chaos, defined on a dynamical level and arising from collisions between molecules, and diffusion, characterized by a mean square displacement proportional to the time. We use a number of…
We examine the destruction of superconducting pairing in metallic grains as their size is decreased for both even and odd numbers of electrons. This occurs when the average level spacing d is of the same order as the BCS order parameter.…
At finite temperature, the field along a linear stretch of correlation length size is supposed to trace the shortest path in the field space given the two end point values, known as the Geodesic rule. In this study, we compute the…
Random hyperspherical harmonics are Gaussian Laplace eigenfunctions on the unit $d$-sphere ($d\ge 2$). We investigate the distribution of their defect i.e., the difference between the measure of positive and negative regions. Marinucci and…
We study percolation problems of overlapping objects where the underlying geometry is such that in D-dimensions, a subset of the directions has a lattice structure, while the remaining directions have a continuum structure. The resulting…
The present work aims at providing a systematic analysis of the current density versus momentum characteristics for a fermionic superfluid throughout the BCS-BEC crossover, even in the fully homogeneous case. At low temperatures, where…