128 research outputs found
Random discrete concave functions on an equilateral lattice with periodic Hessians
Motivated by connections to random matrices, Littlewood-Richardson
coefficients and tilings, we study random discrete concave functions on an
equilateral lattice. We show that such functions having a periodic Hessian of a
fixed average value concentrate around a quadratic function. We consider
the set of all concave functions on an equilateral lattice that
when shifted by an element of have a periodic discrete Hessian,
with period . We add a convex quadratic of Hessian ; the sum is
then periodic with period , and view this as a mean zero function
on the set of vertices of a torus whose
Hessian is dominated by . The resulting set of semiconcave functions forms a
convex polytope . The diameter of is bounded
below by , where is a positive constant depending only on .
Our main result is that under certain conditions, that are met for example when
, for any we have if
is sampled from the uniform measure on . Each
corresponds to a kind of honeycomb. We obtain concentration results for these
as well.Comment: 56 pages. arXiv admin note: substantial text overlap with
arXiv:1909.0858
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