Rank one HCIZ at high temperature: interpolating between classical and free convolutions

We study the rank one Harish-Chandra-Itzykson-Zuber integral in the limit where \frac{N\beta}{2} \to cNβ2→c, called the high-temperature regime and show that it can be used to construct a promising one-parameter interpolation family, with parameter c between the classical and the free convolution. This c-convolution has a simple interpretation in terms of another associated family of distribution indexed by c, called the Markov-Krein transform: the c-convolution of two distributions corresponds to the classical convolution of their Markov-Krein transforms. We derive first cumulant-moment relations, a central limit theorem, a Poisson limit theorem and show several numerical examples of c-convoluted distributions.

Two poisson limit theorems for the coupon collector’s problem with group drawings

In the collector’s problem with group drawings, s out of n different types of coupon are sampled with replacement. In the uniform case, each s-subset of the types has the same probability of being sampled. For this case, we derive a Poisson limit theorem for the number of types that are sampled at most $c-1$ times, where $c \ge 1$ is fixed. In a specified approximate nonuniform setting, we prove a Poisson limit theorem for the special case $c=1$ . As corollaries, we obtain limit distributions for the waiting time for c complete series of types in the uniform case and a single complete series in the approximate nonuniform case.

Poisson Quantum Information

By taking a Poisson limit for a sequence of rare quantum objects, I derive simple formulas for the Uhlmann fidelity, the quantum Chernoff quantity, the relative entropy, and the Helstrom information. I also present analogous formulas in classical information theory for a Poisson model. An operator called the intensity operator emerges as the central quantity in the formalism to describe Poisson states. It behaves like a density operator but is unnormalized. The formulas in terms of the intensity operators not only resemble the general formulas in terms of the density operators, but also coincide with some existing definitions of divergences between unnormalized positive-semidefinite matrices. Furthermore, I show that the effects of certain channels on Poisson states can be described by simple maps for the intensity operators.

Poisson limit of bumping routes in the Robinson–Schensted correspondence

We consider the Robinson–Schensted–Knuth algorithm applied to a random input and investigate the shape of the bumping route (in the vicinity of the y-axis) when a specified number is inserted into a large Plancherel-distributed random tableau. We show that after a projective change of the coordinate system the bumping route converges in distribution to the Poisson process.

Trees grown under young-age preferential attachment

We introduce a class of non-uniform random recursive trees grown with an attachment preference for young age. Via the Chen–Stein method of Poisson approximation, we find that the outdegree of a node is characterized in the limit by ‘perturbed’ Poisson laws, and the perturbation diminishes as the node index increases. As the perturbation is attenuated, a pure Poisson limit ultimately emerges in later phases. Moreover, we derive asymptotics for the proportion of leaves and show that the limiting fraction is less than one half. Finally, we study the insertion depth in a random tree in this class. For the insertion depth, we find the exact probability distribution, involving Stirling numbers, and consequently we find the exact and asymptotic mean and variance. Under appropriate normalization, we derive a concentration law and a limiting normal distribution. Some of these results contrast with their counterparts in the uniform attachment model, and some are similar.

Jet-printing microfluidic devices on demand

ABSTRACTThere is an unmet demand for microfluidics in biomedicine. We describe contactless fabrication of microfluidic circuits on standard Petri dishes using just a dispensing needle, syringe pump, 3-way traverse, cell-culture media, and an immiscible fluorocarbon (FC40). A submerged micro-jet of FC40 is projected through FC40 and media on to the bottom of a dish, where it washes media away to leave liquid fluorocarbon walls pinned to the substrate by interfacial forces. Such fluid walls can be built into almost any imaginable 2D circuit in minutes, which we exploit to clone cells using limiting dilution in a way that beats the Poisson limit, sub-culture adherent cells, and feed arrays of cells continuously for a week. This general method should have wide application in biomedicine.One sentence summaryIn the everyday world, we cannot build complex structures out of liquids as they collapse into puddles; in the microworld we can.

Limit theorems for process-level Betti numbers for sparse and critical regimes

The objective of this study is to examine the asymptotic behavior of Betti numbers of Čech complexes treated as stochastic processes and formed from random points in the d-dimensional Euclidean space ${\mathbb{R}}^d$ . We consider the case where the points of the Čech complex are generated by a Poisson process with intensity nf for a probability density f. We look at the cases where the behavior of the connectivity radius of the Čech complex causes simplices of dimension greater than $k+1$ to vanish in probability, the so-called sparse regime, as well when the connectivity radius is of the order of $n^{-1/d}$ , the critical regime. We establish limit theorems in the aforementioned regimes: central limit theorems for the sparse and critical regimes, and a Poisson limit theorem for the sparse regime. When the connectivity radius of the Čech complex is $o(n^{-1/d})$ , i.e. the sparse regime, we can decompose the limiting processes into a time-changed Brownian motion or a time-changed homogeneous Poisson process respectively. In the critical regime, the limiting process is a centered Gaussian process but has a much more complicated representation, because the Čech complex becomes highly connected with many topological holes of any dimension.