Limit theorems for vertex-reinforced jump processes on regular trees.
For a double channel Markovian queue with finite waiting space and unequal service rates at the two counters, the difference equations satisfied by the Laplace transforms of the state probabilities at finite time are solved and the state probabilities have been obtained. The closed form of the state probabilities can be used to obtain the important parameters of the system.
We consider the ensemble of curves {γα, N: α∈(0, 1], N∈ℕ} obtained by linearly interpolating the values of the normalized theta sum N−1/2∑n=0N'−1exp(πin2α), 0≤N'<N. We prove the existence of limiting finite-dimensional distributions for such curves as N→∞, when α is distributed according to any probability measure λ, absolutely continuous w.r.t. the Lebesgue measure on [0, 1]. Our Main Theorem generalizes a result by Marklof [Duke Math. J.97 (1999) 127–153] and Jurkat and van Horne [Duke...
We study a Markov process on a system of interlacing particles. At large times the particles fill a domain that depends on a parameter ε > 0. The domain has two cusps, one pointing up and one pointing down. In the limit ε ↓ 0 the cusps touch, thus forming a tacnode. The main result of the paper is a derivation of the local correlation kernel around the tacnode in the transition regime ε ↓ 0. We also prove that the local process interpolates between the Pearcey process and the GUE minor process....
Random interlacements at level is a one parameter family of connected random subsets of , (Ann. Math.171(2010) 2039–2087). Its complement, the vacant set at level , exhibits a non-trivial percolation phase transition in (Comm. Pure Appl. Math.62 (2009) 831–858; Ann. Math.171 (2010) 2039–2087), and the infinite connected component, when it exists, is almost surely unique (Ann. Appl. Probab.19(2009) 454–466). In this paper we study local percolative properties of the vacant set of random interlacements...