A phase transition behavior for Brownian motions interacting through their ranks

Consider a time-varying collection of n points on the positive real axis, modeled as Exponentials of n Brownian motions whose drift vector at every time point is determined by the relative ranks of the coordinate processes at that time. If at each time point we divide the points by their sum, under suitable assumptions the rescaled point process converges to a stationary distribution (depending on n and the vector of drifts) as time goes to infinity. This stationary distribution can be exactly computed using a recent result of Pal and Pitman. The model and the rescaled point process are both central objects of study in models of equity markets introduced by Banner, Fernholz, and Karatzas. In this paper, we look at the behavior of this point process under the stationary measure as n tends to infinity. Under a certain ‘continuity at the edge’ condition on the drifts, we show that one of the following must happen: either (i) all points converge to 0, or (ii) the maximum goes to 1 and the rest go to 0, or (iii) the processes converge in law to a non-trivial Poisson–Dirichlet distribution. The underlying idea of the proof is inspired by Talagrand’s analysis of the low temperature phase of Derrida’s Random Energy Model of spin glasses. The main result establishes a universality property for the BFK models and aids in explicit asymptotic computations using known results about the Poisson–Dirichlet law.     

Random overlap structures: properties and applications to spin glasses

Random overlap structures (ROSt’s) are random elements on the space of probability measures on the unit ball of a Hilbert space, where two measures are identified if they differ by an isometry. In spin glasses, they arise as natural limits of Gibbs measures under the appropriate algebra of functions. We prove that the so called ‘cavity mapping’ on the space of ROSt’s is continuous, leading to a proof of the stochastic stability conjecture for the limiting Gibbs measures of a large class of spin glass models. Similar arguments yield the proofs of a number of other properties of ROSt’s that may be useful in future attempts at proving the ultrametricity conjecture. Lastly, assuming that the ultrametricity conjecture holds, the setup yields a constructive proof of the Parisi formula for the free energy of the Sherrington–Kirkpatrick model by making rigorous a heuristic of Aizenman, Sims and Starr.     

Central limit theorem for first-passage percolation time across thin cylinders

We prove that first-passage percolation times across thin cylinders of the form [0, n] × [?h _n , h _n ] ^d-1 obey Gaussian central limit theorems as long as h _n grows slower than n ^1/(d+1). It is an open question as to what is the fastest that h _n can grow so that a Gaussian CLT still holds. Under the natural but unproven assumption about existence of fluctuation and transversal exponents, and strict convexity of the limiting shape in the direction of (1, 0, . . . , 0), we prove that in dimensions 2 and 3 the CLT holds all the way up to the height of the unrestricted geodesic. We also provide some numerical evidence in support of the conjecture in dimension 2.     

Unconditionally-secure key pre-distribution for triangular grid based wireless sensor network

We present an unconditionally-secure key pre-distribution scheme for a wireless sensor network using t-degree bivariate polynomials. The proposed scheme is proven to be perfectly resilient against both node disconnection and link failure. The memory requirements, computation and communication overheads of our scheme are also favorable. Our scheme demonstrates superior performance compared to the existing similar schemes.     

Fock space multireference coupled cluster theory: Study of shape resonance

The complex absorbing potential along with correlated independent particle potential (CIP) Fock space multireference coupled cluster method is used for the study of resonances. We have studied shape resonance of e^?‐ F₂, e^?‐ N₂O and e^?‐CO molecules. In particular, we have studied e^?‐ F₂ scattering at different bond lengths to know whether is bound at the equilibrium bond length of F₂. © 2013 Wiley Periodicals, Inc.