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Asympototic Statistics of Nodal Domains in Quantum Chaotic Billiards
in the Semiclassical Limit

* *Kyle Konrad '12

CS-math major, Dartmouth College

Quantum chaos concerns eigenfunctions of the Laplace operator in a domain
where a billiard ball would bounce chaotically. Such chaotic eigenfunctions,
have been conjectured to share statistical properties of their nodal domains
with a simple percolation model, from which many interesting quantities can
be computed analytically. We numerically test conjectures on the number and
size of nodal domains of quantum chaotic eigenfunctions at very high
energies, approaching the semiclassical limit. We use a highly efficient
scaling method to quickly compute eigenfunctions at low resolution and
interpolate to higher resolution. We collect around 100,000 eigenfunctions
and count 1.5 billion nodal domains. Our results provide strong numerical
evidence for conjectures on the mean number and size of nodal domains but
disagree with the conjectured variance.

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