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Hana Kogan

Publications and source records attributed to Hana Kogan.

6 recordsLinked to original sources
arXiv2014

Coarsening dynamics on $\mathbb{Z}^d$ with frozen vertices

We study Markov processes in which $\pm 1$-valued random variables $σ_x(t), x\in \mathbb{Z}^d$, update by taking the value of a majority of their nearest neighbors or else tossing a fair coin in case of a tie. In the presence of a random environment of frozen plus (resp., minus) vertices with density $ρ^+$ (resp., $ρ^-$), we study the prevalence of vertices that are (eventually) fixed plus or fixed minus or flippers (changing forever). Our main results are that, for $ρ^+ >0$ and $ρ^- =0$, all sites are fixed plus, while for $ρ^+ >0$ and $ρ^-$ very small (compared to $ρ^+$), the fixed minus and flippers together do not percolate. We also obtain some results for deterministic placement of frozen vertices.

math.PR↗
arXiv2013

Fixation for coarsening dynamics in 2D slabs

For the zero temperature limit of Ising Glauber Dynamics on 2D slabs the existence or nonexistence of vertices that do not fixate is determined as a function of slab thickness.

math.PR↗
arXiv2013

Coarsening in 2D slabs

We study coarsening; that is, the zero-temperature limit of Glauber dynamics in the standard Ising model on slabs S_k = Z^2 x {0, ..., k-1} of all thicknesses k \geq 2 (with free and periodic boundary conditions in the third coordinate). We show that with free boundary conditions, for k \geq 3, some sites fixate for large times and some do not, whereas for k=2, all sites fixate. With periodic boundary conditions, for k \geq 4, some sites fixate and others do not, while for k=2 and 3, all sites fixate.

math.PR↗
arXiv2011

Permanental Vectors

A permanental vector is a generalization of a vector with components that are squares of the components of a Gaussian vector, in the sense that the matrix that appears in the Laplace transform of the vector of Gaussian squares is not required to be either symmetric or positive definite. In addition the power of the determinant in the Laplace transform of the vector of Gaussian squares, which is -1/2, is allowed to be any number less than zero. It was not at all clear what vectors are permanental vectors. In this paper we characterize all permanental vectors in $R^{3}_{+}$ and give applications to permanental vectors in $R^{n}_{+}$ and to the study of permanental processes.

math.PR↗
arXiv2010

Permanental Processes

This is a survey of results about permanental processes, real valued positive processes which are a generalization of squares of Gaussian processes. In a certain sense the symmetric positive definite function that determines a Gaussian process is replaced by a function that is not necessarily symmetric nor positive definite, but that nevertheless determines a stochastic process. This is a new avenue of research with very many open problems.

math.PR↗