arXiv subjects
Publications and source records attributed to Rahul Roy.
A square lattice model which exhibits a nonzero quantized Hall conductance in a zero net magnetic field at certain values of the parameters is presented. The quantization is due to the existence of a topological winding number that characterizes the quantum Hall phases and is expected to survive the effects of weak disorder.
We analyze the threshold network model in which a pair of vertices with random weights are connected by an edge when the summation of the weights exceeds a threshold. We prove some convergence theorems and central limit theorems on the vertex degree, degree correlation, and the number of prescribed subgraphs. We also generalize some results in the spatially extended cases.
Consider the d-dimensional lattice Z^d where each vertex is ``open'' or ``closed'' with probability p or 1-p, respectively. An open vertex v is connected by an edge to the closest open vertex w such that the dth co-ordinates of v and w satisfy w(d)=v(d)-1. In case of nonuniqueness of such a vertex w, we choose any one of the closest vertices with equal probability and independently of the other random mechanisms. It is shown that this random graph is a tree almost surely for d=2 and 3 and it is an infinite collection of distinct trees for d\geq4. In addition, for any dimension, we show that there is no bi-infinite path in the tree and we also obtain central limit theorems of (a) the number of vertices of a fixed degree νand (b) the number of edges of a fixed length l.
Sticks at one of different orientation are placed in an i.i.d. fashion at points of a Poisson point process of intensity $λ$. Sticks of the same direction have the same length, while sticks in different directions may have different lengths. We study the geometry of finite cluster as $λ\to \infty$. The asymptotic shape of the custer being determined by the probabilities of the sticks in various direction and their lengths and orientations. We also obtain the limiting geometric structure of this component.
The excitation spectrum of a two-dimensional $p_x+ip_y$ fermionic superfluid, such as a thin film of $^3$He-A, includes a gapless Majorana-Weyl fermion which is confined to the boundary by Andreev reflection. There is also a persistent ground-state boundary current which provides a droplet containing $N$ particles with angular momentum $\hbar N/2$. Both of these boundary effects are associated with bulk Chern-Simons effective actions. We show that the gapless edge-mode is required for the gauge invariance of the total effective action, but the same is not true of the boundary current.