9,579 research outputs found
Interpolation between Airy and Poisson statistics for unitary chiral non-Hermitian random matrix ensembles
We consider a family of chiral non-Hermitian Gaussian random matrices in the unitarily invariant symmetry class. The eigenvalue distribution in this model is expressed in terms of Laguerre polynomials in the complex plane. These are orthogonal with respect to a non-Gaussian weight including a modified Bessel function of the second kind, and we give an elementary proof for this. In the large n limit, the eigenvalue statistics at the spectral edge close to the real axis are described by the same family of kernels interpolating between Airy and Poisson that was recently found by one of the authors for the elliptic Ginibre ensemble. We conclude that this scaling limit is universal, appearing for two different non-Hermitian random matrix ensembles with unitary symmetry. As a second result we give an equivalent form for the interpolating Airy kernel in terms of a single real integral, similar to representations for the asymptotic kernel in the bulk and at the hard edge of the spectrum. This makes its structure as a one-parameter deformation of the Airy kernel more transparent
Interpolating Between Hilbert-Samuel and Hilbert-Kunz Multiplicity
We define a function, called s-multiplicity, that interpolates between
Hilbert-Samuel multiplicity and Hilbert-Kunz multiplicity by comparing powers
of ideals to the Frobenius powers of ideals. The function is continuous in s,
and its value is equal to Hilbert-Samuel multiplicity for small values of s and
is equal to Hilbert-Kunz multiplicity for large values of s. We prove that it
has an Associativity Formula generalizing the Associativity Formulas for
Hilbert-Samuel and Hilbert-Kunz multiplicity. We also define a family of
closures such that if two ideals have the same s-closure then they have the
same s-multiplicity, and the converse holds under mild conditions. We describe
the s-multiplicity of monomial ideals in toric rings as a certain volume in
real spaceComment: 19 page
Interpolation in local theory extensions
In this paper we study interpolation in local extensions of a base theory. We
identify situations in which it is possible to obtain interpolants in a
hierarchical manner, by using a prover and a procedure for generating
interpolants in the base theory as black-boxes. We present several examples of
theory extensions in which interpolants can be computed this way, and discuss
applications in verification, knowledge representation, and modular reasoning
in combinations of local theories.Comment: 31 pages, 1 figur
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