Balls-in-boxes duality for coalescing random walks and coalescing Brownian motions

We present a duality relation between two systems of coalescing random walks and an analogous duality relation between two systems of coalescing Brownian motions. Our results extends previous work in the literature and we apply it to the study of a system of coalescing Brownian motions with Poisson immigration.Comment: 13 page

Coalescing at 8 GeV in the Fermilab Main Injector

For Project X, it is planned to inject a beam of 3 10**11 particles per bunch into the Main Injector. To prepare for this by studying the effects of higher intensity bunches in the Main Injector it is necessary to perform coalescing at 8 GeV. The results of a series of experiments and simulations of 8 GeV coalescing are presented. To increase the coalescing efficiency adiabatic reduction of the 53 MHz RF is required, resulting in ~70% coalescing efficiency of 5 initial bunches. Data using wall current monitors has been taken to compare previous work and new simulations for 53 MHz RF reduction, bunch rotations and coalescing, good agreement between experiment and simulation was found. Possible schemes to increase the coalescing efficiency and generate even higher intensity bunches are discussed. These require improving the timing resolution of the low level RF and/or tuning the adiabatic voltage reduction of the 53 MHz.Comment: 3 pp. 3rd International Particle Accelerator Conference (IPAC 2012) 20-25 May 2012, New Orleans, Louisian

A superprocess involving both branching and coalescing

We consider a superprocess with coalescing Brownian spatial motion. We first prove a dual relationship between two systems of coalescing Brownian motions. In consequence we can express the Laplace functionals for the superprocess in terms of coalescing Brownian motions, which allows us to obtain some explicit results. We also point out several connections between such a superprocess and the Arratia flow. A more general model is discussed at the end of this paper.Comment: 25 page

Coalescing Brownian flows: A new approach

The coalescing Brownian flow on $\mathbb{R}$ is a process which was introduced by Arratia [Coalescing Brownian motions on the line (1979) Univ. Wisconsin, Madison] and T\'{o}th and Werner [Probab. Theory Related Fields 111 (1998) 375-452], and which formally corresponds to starting coalescing Brownian motions from every space-time point. We provide a new state space and topology for this process and obtain an invariance principle for coalescing random walks. This result holds under a finite variance assumption and is thus optimal. In previous works by Fontes et al. [Ann. Probab. 32 (2004) 2857-2883], Newman et al. [Electron. J. Probab. 10 (2005) 21-60], the topology and state-space required a moment of order $3-\varepsilon$ for this convergence to hold. The proof relies crucially on recent work of Schramm and Smirnov on scaling limits of critical percolation in the plane. Our approach is sufficiently simple that we can handle substantially more complicated coalescing flows with little extra work - in particular similar results are obtained in the case of coalescing Brownian motions on the Sierpinski gasket. This is the first such result where the limiting paths do not enjoy the noncrossing property.Comment: Published at http://dx.doi.org/10.1214/14-AOP957 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Weak convergence of the localized disturbance flow to the coalescing Brownian flow

We define a new state-space for the coalescing Brownian flow, also known as the Brownian web, on the circle. The elements of this space are families of order-preserving maps of the circle, depending continuously on two time parameters and having a certain weak flow property. The space is equipped with a complete separable metric. A larger state-space, allowing jumps in time, is also introduced, and equipped with a Skorokhod-type metric, also complete and separable. We prove that the coalescing Brownian flow is the weak limit in this larger space of a family of flows which evolve by jumps, each jump arising from a small localized disturbance of the circle. A local version of this result is also obtained, in which the weak limit law is that of the coalescing Brownian flow on the line. Our set-up is well adapted to time-reversal and our weak limit result provides a new proof of time-reversibility of the coalescing Brownian flow. We also identify a martingale associated with the coalescing Brownian flow on the circle and use this to make a direct calculation of the Laplace transform of the time to complete coalescence.Comment: Published at http://dx.doi.org/10.1214/13-AOP845 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org). arXiv admin note: substantial text overlap with arXiv:0810.021