Let
Z^{d }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
?Z^{d}) is introduced. For all z,w ?Z^{d}, if z = w then
), where _{st. }denotes
stochastic domination. The basic branching random walk (BRW) on Z^{d }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 A^{0 }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.
A^{0}. 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 A^{0}. 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 Z^{d }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) ? Z^{2}, 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 A^{0 }above and, remark that it will be seen
by the proof that extensions for the process on Z^{d }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
L^{0 }denote L shifted to odd coordinates instead. Let further T_{n }be the subset of L^{0 }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, S_{n }:= {0}×{0,1,...,n}
is an axis of symmetry. By symmetry of points about S_{n }each point (-y,k),
y = 1, has a unique counterpart
mirror image (y,k), while points of S_{n }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 T_{n}.
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 Z^{d}, MDAD for sites on one axis is
shown in [6]. For a rigorous proof of the MDAD for the BRW on Z^{d}, 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).