134 research outputs found
High-Dimensional Lipschitz Functions are Typically Flat
A homomorphism height function on the -dimensional torus
is a function taking integer values on the vertices of the torus with
consecutive integers assigned to adjacent vertices. A Lipschitz height function
is defined similarly but may also take equal values on adjacent vertices. In
each model, we consider the uniform distribution over such functions, subject
to boundary conditions. We prove that in high dimensions, with zero boundary
values, a typical function is very flat, having bounded variance at any fixed
vertex and taking at most values with high probability. Our
results extend to any dimension , if is replaced by an
enhanced version of it, the torus for
some fixed . This establishes one side of a conjectured roughening
transition in dimensions. The full transition is established for a class of
tori with non-equal side lengths. We also find that when is taken to
infinity while remains fixed, a typical function takes at most values
with high probability, where for the homomorphism model and for the
Lipschitz model. Suitable generalizations are obtained when grows with .
Our results apply also to the related model of uniform 3-coloring and
establish, for certain boundary conditions, that a uniformly sampled proper
3-coloring of will be nearly constant on either the even or
odd sub-lattice.
Our proofs are based on a combinatorial transformation and on a careful
analysis of the properties of a class of cutsets which we term odd cutsets. For
the Lipschitz model, our results rely also on a bijection of Yadin. This work
generalizes results of Galvin and Kahn, refutes a conjecture of Benjamini,
Yadin and Yehudayoff and answers a question of Benjamini, H\"aggstr\"om and
Mossel.Comment: 63 pages, 5 figures (containing 10 images). Improved introduction and
layout. Minor correction
On rough isometries of Poisson processes on the line
Intuitively, two metric spaces are rough isometric (or quasi-isometric) if
their large-scale metric structure is the same, ignoring fine details. This
concept has proven fundamental in the geometric study of groups. Ab\'{e}rt, and
later Szegedy and Benjamini, have posed several probabilistic questions
concerning this concept. In this article, we consider one of the simplest of
these: are two independent Poisson point processes on the line rough isometric
almost surely? Szegedy conjectured that the answer is positive. Benjamini
proposed to consider a quantitative version which roughly states the following:
given two independent percolations on , for which constants are
the first points of the first percolation rough isometric to an initial
segment of the second, with the first point mapping to the first point and with
probability uniformly bounded from below? We prove that the original question
is equivalent to proving that absolute constants are possible in this
quantitative version. We then make some progress toward the conjecture by
showing that constants of order suffice in the quantitative
version. This is the first result to improve upon the trivial construction
which has constants of order . Furthermore, the rough isometry we
construct is (weakly) monotone and we include a discussion of monotone rough
isometries, their properties and an interesting lattice structure inherent in
them.Comment: Published in at http://dx.doi.org/10.1214/09-AAP624 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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