AN ASPECT OF PARTICLES SPATIAL COMPETITION
Achillefs Tzioufas
Instituto de Matematica e Estatstica, Universidade de S~ao Paulo Rua do Mat~ao, 1010, CEP 05508-900- S~ao Paulo, Brasiltzioufas@ime.usp.br
Keywords: Stochastic domination; symmetry; monotonicity; spatial branching processes; oriented percolation
Abstract

The branching random walk on integer lattices is shown to be monotone decreasing in ascending direction (MDAD) as a corollary of an observation in regard to the lineage of particles from antisymmetric initial states, and a related property of percolation paths on the usual oriented lattices is also given.

Article Information

Identifiers and Pagination:
Year:2016
Volume:1
First Page:5
Last Page:7
Publisher Id:.1:1.2016
Article History:
Received:December 16, 2016
Accepted:December 23, 2016
Collection year:2016
First Published:December 27, 2016

Let Zd be the set of all d-vectors of integers endowed with the natural preorder induced by coordinate-wise modulus, denoted by =. The following criterion for a collection X of random variables (X(z);z ?Zd) is introduced. For all z,w ?Zd, if z = w then ), where st. denotes stochastic domination. The basic branching random walk (BRW) on Zd started from a particle at the origin is shown to satisfy this criterion at all times. For terminology and background on the BRW we refer the reader to I.1 in [5].

The method of proof here is by deduction from a spatial competition property for the discrete-time counterpart of the process done in §1, whereas, in §2, a property of that type for oriented percolation paths is shown.

1. a) A population of particles inhabiting Z, the set of sites positioned at the integers, evolves in generations in discrete-time according to the following, parameter p, rules. Every particle begets another at each nearest site with probability p at the instant of its demise. All births are independent of one another.

Started from one progenitor at the origin, {0}, note that the only feasible (inhabited with non-zero probability) sites are the even at odd times and the odd at even times.

Proposition 1. At all times for any duad of nearest feasible sites the cardinality of particles at the one nearest to the origin stochastically dominates the other.

The Proposition above comes from translation invariance and taking C in the next statement to be singletons.

A. At all time and on any subset C of {0,1,2,...} the distribution of particles descending from {-1} equals that of those descending from {1} and visited {0} at some time prior or equal to that time.

The statement in A is a consequence of that in A0 below by noting that particles with non-negative spatial coordinate at any given time have to have visited at least once site {0} at some time prior or equal to that time.

A0. At all time the distribution of particles descending from {1} and visited site {0} prior to that time equals that of those from {-1} instead.

We elaborate on the idea of ”switching the spin upon collisions at the symmetry axis” coupling, thus, proving A0. Particles descending from {-1} are tagged as a-particles for as long as they occupy sites with negative spatial coordinate, while those descending from {1} are tagged as ß-particles for as long as they occupy sites with positive one. Further, particles born at {0} from a-particles are tagged as ?-particles, while those born from ß-particles are tagged as d-particles. Coupling a-particles with ß-particles antithetically imposes that births of ?-particles and dparticles are simultaneous, which thus permits coupling them upon birth identically.

From the Proposition above the result claimed in the introduction for d = 1 follows by drawing upon known approximation methods, on which we suffice it to note that the one proposed in [2] is used here in an essential way. Further, a little thought shows that the Proposition above is extended for the process corresponding to a) on Zd analogously. From this the result claimed follows in all generality since for each d the extension of the method refer to is carried out straightforwardly.

Finally, it is worth remarking here that the resultant dismantling from consideration of the discrete time counterpart of the process had been instructive throughout this section.

2. Let L be the usual oriented percolation lattice, that is, the set of space-time points (y,k) ? Z2, y + k even, k = 0, with adjunct points, (y - 1,k + 1) and (y + 1,k + 1). By bond percolation retaining parameter p it is understood that every bond is declared independently open with probability p, and as usual a path is defined as a sequence of consecutive adjunct points that includes exactly one at each time. The next statement is better illustrated by considering the embedded process: b) The population of particles evolves on Z as in a), but births from the same site to each nearest site are identically dependent. To see the connection note that the cardinality of open paths to a site in the former is the number of particles in the latter for appropriately similar initial conditions.

Note that the next statement is in the spirit of that in A0 above and, remark that it will be seen by the proof that extensions for the process on Zd analogously hold.

B. At all time n the distribution of particles descending from {1} and visited site {0} at any fixed time m, m = n, equals that of those from {-1} instead.

Let L0 denote L shifted to odd coordinates instead. Let further Tn be the subset of L0 whose points are delimited within the isosceles trapezoid, vertices the points (-1,0), (1,0) and (-1 - n,n), (1 + n,n), for which, note that, Sn := {0}×{0,1,...,n} is an axis of symmetry. By symmetry of points about Sn each point (-y,k), y = 1, has a unique counterpart mirror image (y,k), while points of Sn we regard as the mirror image of themselves. Accordingly, the mirror image of every pair of neighbouring points is regarded to be the pair of their mirroring points.

The following idea of ”rotating the sample point” coupling proves B. Consider a realization of bond percolation retaining parameter p on Tn. Consider the operation of interchanging the value of the Bernoulli r.v. assigned to each pair of adjunct points with that associated to the pair’s mirror image prior or equal to time m. Note that the operation results in dissection of paths passing through (0,m) at that point as well as in their reflection about the symmetry axis prior to that time. Since further this operation defines a one-to-one correspondence and preserves the total number of bonds present the proof is completed.

Postscript. Literature related to the MDAD property is as follows. For the basic contact process started from the origin on Z, this is shown in [1], see also [2, 3]. Regarding highly subcritical bond percolation on Zd, MDAD for sites on one axis is shown in [6]. For a rigorous proof of the MDAD for the BRW on Zd, see [4], Lemma 11; Proposition 1 was shown independently by the author in [7].


References

1.       Andjel, E. D., and L. F. Gray. ”Extreme paths in oriented two-dimensional percolation.” Journal of Applied Probability 53.2 (2016): 369-380.

2.       Andjel, E. and Sued, M. An Inequality for Oriented 2-D Percolation. In and Out of Equilibrium 2: 21-30. (2008)

3.       Gray, L. Is the contact process dead? In Proceedings of the 1989 AMS Seminar on Random Media, vol. 27, pp. 19-29. (1991)

4.       Lalley, S. P., and Zheng, X. (2011). Occupation statistics of critical branching random walks in two or higher dimensions. The Annals of Probability, 39(1), 327-368.

5.       Liggett, T. Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Springer, New York. (1999)

6.       Lima, B. N., Procacci, A., and Sanchis, R. (2015). A remark on monotonicity in

7.       Bernoulli bond Percolation. Journal of Statistical Physics, 160(5), 1244-1248.

8.       Tzioufas, A. An aspect of particles’ spatial competition. arXiv:1303.4093 (2013).


© 2016 The Author(s). This open access article is distributed under a Creative Commons Attribution (CC-BY) 4.0 license. You are free to: Share — copy and redistribute the material in any medium or format Adapt — remix, transform, and build upon the material for any purpose, even commercially. The licensor cannot revoke these freedoms as long as you follow the license terms. Under the following terms: Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. No additional restrictions You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits
Editor in Chief
Zhilin Li (Ph.D. (Applied Mathematics))
Professor of Mathematics, Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205 USA

Bibliography

Prof. Dr. Zhilin Li is associated with Department of Mathematics, North Carolina State University, US since 1997. He was graduated from Nanking Normal University (Mathematics B.S.) in 1982. Whereas, he has completed his Master in Mathematics from University of Washington Applied in 1991. The University of Washington awarded him PhD in Applied Mathematics in 1994. He has published 127 quality manuscript since October 31, 2016. Moreover, he received the Research Grants Current and past from NSF, NIH, NSF/NIGMS, ARO, AFOSR, Oak Ridge, DOE/ARO etc. He is collaborated and affiliated with Juan Alvares, UAH, Spain; Philippe Angot, Aix-Marceille University, France; J. Chen & H. Ji, NNU, China; K Ito, SR Lubkin, NCSU; M-C Lai, Taiwan; R. Luo, UCI; J. Xia, Purdue; Hayk Mikayelyan, Nottingham. Moreover, he has advised and supervised more than 15 graduated students to complete their research projects.

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