10 research outputs found
Two-Point Correlation Functions and Universality for the Zeros of Systems of SO(n+1)-invariant Gaussian Random Polynomials
We study the two-point correlation functions for the zeroes of systems of
-invariant Gaussian random polynomials on and systems
of -invariant Gaussian analytic functions. Our result
reflects the same "repelling," "neutral," and "attracting" short-distance
asymptotic behavior, depending on the dimension, as was discovered in the
complex case by Bleher, Shiffman, and Zelditch. For systems of the -invariant Gaussian analytic functions we also obtain a
fast decay of correlations at long distances.
We then prove that the correlation function for the -invariant Gaussian analytic functions is "universal,"
describing the scaling limit of the correlation function for the restriction of
systems of the -invariant Gaussian random polynomials to any
-dimensional submanifold . This provides a
real counterpart to the universality results that were proved in the complex
case by Bleher, Shiffman, and Zelditch. (Our techniques also apply to the
complex case, proving a special case of the universality results of Bleher,
Shiffman, and Zelditch.)Comment: 28 pages, 1 figure. To appear in International Mathematics Research
Notices (IMRN
On the Number of 2-Protected Nodes in Tries and Suffix Trees
We use probabilistic and combinatorial tools on strings to discover the average number of 2-protected nodes in tries and in suffix trees. Our analysis covers both the uniform and non-uniform cases. For instance, in a uniform trie with leaves, the number of 2-protected nodes is approximately 0.803, plus small first-order fluctuations. The 2-protected nodes are an emerging way to distinguish the interior of a tree from the fringe
On the Number of 2-Protected Nodes in Tries and Suffix Trees
We use probabilistic and combinatorial tools on strings to discover the average number of 2-protected nodes in tries and in suffix trees. Our analysis covers both the uniform and non-uniform cases. For instance, in a uniform trie with leaves, the number of 2-protected nodes is approximately 0.803, plus small first-order fluctuations. The 2-protected nodes are an emerging way to distinguish the interior of a tree from the fringe