2020.05.14 Jack Hanson -  Critical first-passage percolation (part 2)
Channel: One World Probability
Date: 2020-05-14

Part 1: background and behaviour on regular trees 
Part 2: limit theorems for lattice first-passage times 


For many lattice models in probability, the high-dimensional behaviour is
well-predicted by the behaviour of a corresponding random model defined on a
regular tree. Rigorous results of this kind have been proved for self-avoiding
walk, Bernoulli percolation, and the Ising model. One model where no such
results are known is first-passage percolation, where there is a rather
comprehensive understanding for trees, but relatively little is known in the
lattice setting.

By critical first passage percolation we mean first-passage percolation where
the edge weight distribution mu satisfies mu({0}) = p_c. This creates long
paths with passage time zero, which causes passage times to grow sublinearly.

It turns out that for critical first-passage percolation, it is possible to
prove that high-dimensional lattice models have the same first passage time
asymptotics as tree models. More precisely, for a wide range of critical edge
weight distributions mu, there is an explicit constant C=C(mu) such that the
following holds almost surely.

* If d⋗= 8 then tau_n/(log log n) --⋗ C.

* If d=7 then tau_n/(log log n) --⋗ 3C/2.

In both cases, logs are base-two. The constant C(\mu) is the edge weight
distribution's gap: C(mu) = inf( x ⋗ 0: mu((0,x]) ⋗ 0).

This doubly logarithmic growth is different from the low-dimensional picture;
when d=2, the passage times in critical first-passage percolation instead
satisfy tau_n/log n --⋗ C/(2sqrt{3}*pi). Here the first-passage times grow
singly logarithmically; the constant C(mu) is again the gap.

The first lecture, delivered by Louigi Addario-Berry, will provide more
introductory material related to the above story, covering the following
topics:
1) Critical percolation: behaviour of cluster sizes and diameters in high
dimensions (and on trees).
2) First-passage percolation: a brief overview of known results.
3) Asymptotics for non-critical and critical first-passage percolation on
trees.

The second lecture, delivered by Jack Hanson, will present the aforementioned
results about critical first-passage percolation on the lattice. The behavior
of critical FPP on trees accurately predicts the rate of growth for d ⋗= 8,
and the story from the tree setting provides a reasonable heuristic for the
rate of growth in these settings. Unlike on trees, there is no exact renewal
structure for the growth process on the lattice, and one must ensure that the
growing region is sufficiently spatially "spread out" to regain some
independence. In the case d=7, there is in a sense less independence, and the
growth process slows down. Our techniques also allow us to bound the
probabilities of so-called "defected arm events", which can be thought of as
the probability of an extreme lower-tail event for the critical first-passage
time.