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A radix sort tree arises when storing distinct infinite binary words in the leaves of a binary tree such that for any two words their common prefixes coincide with the common prefixes of the corresponding two leaves. If one deletes the out-degree $1$ vertices in the radix sort tree and closes up the gaps, then the resulting PATRICIA tree maintains all the information that is necessary for sorting the infinite words into lexicographic order. We investigate the PATRICIA chains -- the tree-valued Markov chains that arise when successively building the PATRICIA trees for the collection of infinite binary words $Z_1,ldots, Z_n$, $n=1,2,ldots$, where the source words $Z_1, Z_2,ldots$ are independent and have a common diffuse distribution on ${0,1}^infty$. It turns out that the PATRICIA chains share a common collection of backward transition probabilities and that these are the same as those of a chain introduced by Remy for successively generating uniform random binary trees with larger and larger numbers of leaves. This means that the infinite bridges of any PATRICIA chain (that is, the chains obtained by conditioning a PATRICIA chain on its remote future) coincide with the infinite bridges of the Remy chain. The infinite bridges of the Remy chain are characterized concretely in Evans, Grubel, and Wakolbinger 2017 and we recall that characterization here while adding some details and clarifications.
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Spectral decomposition of the covariance operator is one of the main building blocks in the theory and applications of Gaussian processes. Unfortunately it is notoriously hard to derive in a closed form. In this paper we consider the eigenproblem for Gaussian bridges. Given a {em base} process, its bridge is obtained by conditioning the trajectories to start and terminate at the given points. What can be said about the spectrum of a bridge, given the spectrum of its base process? We show how this question can be answered asymptotically for a family of processes, including the fractional Brownian motion.
In this paper we study fluctuations of extreme particles of nonintersecting Brownian bridges starting from $a_1leq a_2leq cdots leq a_n$ at time $t=0$ and ending at $b_1leq b_2leq cdotsleq b_n$ at time $t=1$, where $mu_{A_n}=(1/n)sum_{i}delta_{a_i}, mu_{B_n}=(1/n)sum_i delta_{b_i}$ are discretization of probability measures $mu_A, mu_B$. Under regularity assumptions of $mu_A, mu_B$, we show as the number of particles $n$ goes to infinity, fluctuations of extreme particles at any time $0<t<1$, after proper rescaling, are asymptotically universal, converging to the Airy point process.
Let $U$ be a Haar distributed matrix in $mathbb U(n)$ or $mathbb O (n)$. In a previous paper, we proved that after centering, the two-parameter process [T^{(n)} (s,t) = sum_{i leq lfloor ns rfloor, j leq lfloor ntrfloor} |U_{ij}|^2] converges in distribution to the bivariate tied-down Brownian bridge. In the present paper, we replace the deterministic truncation of $U$ by a random one, where each row (resp. column) is chosen with probability $s$ (resp. $t$) independently. We prove that the corresponding two-parameter process, after centering and normalization by $n^{-1/2}$ converges to a Gaussian process. On the way we meet other interesting convergences.
We show that the squared maximal height of the top path among $N$ non-intersecting Brownian bridges starting and ending at the origin is distributed as the top eigenvalue of a random matrix drawn from the Laguerre Orthogonal Ensemble. This result can be thought of as a discrete version of K. Johanssons result that the supremum of the Airy$_2$ process minus a parabola has the Tracy-Widom GOE distribution, and as such it provides an explanation for how this distribution arises in models belonging to the KPZ universality class with flat initial data. The result can be recast in terms of the probability that the top curve of the stationary Dyson Brownian motion hits an hyperbolic cosine barrier.
Let U be a Haar distributed unitary matrix in U(n)or O(n). We show that after centering the double index process $$ W^{(n)} (s,t) = sum_{i leq lfloor ns rfloor, j leq lfloor ntrfloor} |U_{ij}|^2 $$ converges in distribution to the bivariate tied-down Brownian bridge. The proof relies on the notion of second order freeness.
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