Convergence of the critical finite-range contact process
to super-Brownian motion above the upper critical dimension:
The higher-point functions
Remco van der Hofstad, Eindhoven University of Technology
Akira Sakai, Hokkaido University
Abstract
In this paper, we investigate the contact process
higher-point functions which denote the probability that
the infection started at the origin at time 0 spreads to
an arbitrary number of other individuals at various later times.
Together with the results of the two-point function in [16],
on which our proofs crucially rely, we prove that the higher-point
functions converge to the moment measures of the canonical measure
of super-Brownian motion above the upper critical dimension 4.
We also prove partial results for in dimension at most 4 in a local mean-field
setting.
The proof is based on the lace expansion for the time-discretized
contact process, which is a version of oriented percolation.
For ordinary oriented percolation, we thus reprove the results
of [20]. The lace expansion coefficients are shown to obey
bounds uniformly in the discretization parameter, which allows
us to establish the scaling results also for the contact process
We also show that the main term of the vertex factor, which is
one of the non-universal constants in the scaling limit, is 1 for
oriented percolation, and 2 for the contact process, while the main
terms of the other non-universal constants are independent of
the discretization parameter.
The lace expansion we develop in this paper is adapted to both the higher-point
functions and the survival probability. This unified approach makes
it easier to relate the expansion coefficients derived in this paper and
the expansion coefficients for the survival probability, which will
be investigated in a future paper [18].
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