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The goal of this paper is to apply the collection of mathematical tools known as the "method of arbitrary functions" to analyze how probability arises from quantum dynamics. We argue that in a toy model of quantum measurement the Born rule…
We study the competing urn model in which $m$ balls are placed independently into $n$ urns according to (possibly distinct) ball distributions. Kahn and Neiman (2010) showed that, under identical ball distributions, the induced urn measure…
In this paper, we consider a multi-drawing urn model with random addition. At each discrete time step, we draw a sample of m balls. According to the composition of the drawn colors, we return the balls together with a random number of balls…
Consider a number, finite or not, of urns each with fixed capacity $r$ and balls randomly distributed among them. An overflow is the number of balls that are assigned to urns that already contain $r$ balls. When $r=1$, using analytic…
Consider an urn model whose replacement matrix is triangular, has all entries nonnegative and the row sums are all equal to one. We obtain the strong laws for the counts of balls corresponding to each color. The scalings for these laws…
We study an urn process with two urns, initialized with a ball each. Balls are added sequentially, the urn being chosen independently with probability proportional to the $\alpha^{th}$ power $(\alpha >1)$ of the existing number of balls. We…
We study the probability distribution function of the ground-state energies of the disordered one-dimensional Ising spin chain with power-law interactions using a combination of parallel tempering Monte Carlo and branch, cut, and price…
Adaptive randomly reinforced urn (ARRU) is a two-color urn model where the updating process is defined by a sequence of non-negative random vectors $\{(D_{1,n}, D_{2,n});n\geq1\}$ and randomly evolving thresholds which utilize accruing…
An urn contains balls of d colors. At each time, a ball is drawn and then replaced together with a random number of balls of the same color. Assuming that some colors are dominated by others, we prove central limit theorems. Some…
We consider an urn model, whose replacement matrix has all entries nonnegative and is balanced, that is, has constant row sums. We obtain the rates of the counts of balls corresponding to each color for the strong laws to hold. The analysis…
For a one-locus haploid infinite population with discrete generations, the celebrated Kingman's model describes the evolution of fitness distributions under the competition of selection and mutation, with a constant mutation probability.…
Recent studies have indicated that the coarse grained dynamics of a large class of traffic models and driven-diffusive systems may be described by urn models. We consider a class of one-dimensional urn models whereby particles hop from an…
We present here two examples of stochastic modelings of social phenomena. The first topic is pedestrian counter flow. Two groups of model pedestrians move in opposite directions and create congestions. It will be shown that this congestion…
For a probability distribution $P$ on an at most countable alphabet $\mathcal A$, this article gives finite sample bounds for the expected occupancy counts $\mathbb E K_{n,r}$ and probabilities $\mathbb E M_{n,r}$. Both upper and lower…
After reviewing the general scaling properties of aging systems, we present a numerical study of the slow evolution induced in the zeta urn model by a quench from a high temperature to a lower one where a condensed equilibrium phase exists.…
We numerically investigate the statistical properties of Wigner delay time in Anderson disordered 1D, 2D and quantum dot (QD) systems. The distribution of proper delay time for each conducting channel is found to be universal in 2D and QD…
We introduce preferential behavior into the study on statistical mechanics of money circulation. The computer simulation results show that the preferential behavior can lead to power laws on distributions over both holding time and amount…
We study an urn process containing red and blue balls and two different strategies to reinforce the urn. Namely, a generalized P\'olya-type strategy versus an i.i.d. one. At each step, one of the two reinforcement strategies is chosen by…
Linearizing the Heisenberg equations of motion around the ground state of an interacting quantum many-body system, one gets a time-evolution generator in the positive cone of a real symplectic Lie algebra. The presence of disorder in the…
Flocking is a prime example of how robust collective behavior can emerge from simple interaction rules. The flocking transition has been studied extensively since the inception of the original Vicsek model. Here, we introduce a novel…