842 research outputs found
A (2+1)-dimensional Gaussian field as fluctuations of quantum random walks on quantum groups
This paper introduces a (2+1)-dimensional Gaussian field which has the
Gaussian free field on the upper half-plane with zero boundary conditions as
certain two-dimensional sections. Along these sections, called space-like
paths, it matches the Gaussian field from eigenvalues of random matrices and
from a growing random surface. However, along time-like paths the behavior is
different.
The Gaussian field arises as the asymptotic fluctuations in quantum random
walks on quantum groups U_q(gl_n). This quantum random walk is a q-deformation
of previously considered quantum random walks. The construction is accomplished
utilizing Etingof-Kirillov difference operators in place of differential
operators on GL(n). When restricted to the space-like paths, the moments of the
quantum random walk match the moments of the growing random surface
Asymptotics of Plancherel measures for the infinite-dimensional unitary group
We study a two-dimensional family of probability measures on infinite
Gelfand-Tsetlin schemes induced by a distinguished family of extreme characters
of the infinite-dimensional unitary group. These measures are unitary group
analogs of the well-known Plancherel measures for symmetric groups. We show
that any measure from our family defines a determinantal point process, and we
prove that in appropriate scaling limits, such processes converge to two
different extensions of the discrete sine process as well as to the extended
Airy and Pearcey processes.Comment: 39 page
Strong Szego asymptotics and zeros of the zeta function
Assuming the Riemann hypothesis, we prove the weak convergence of linear
statistics of the zeros of L-functions towards a Gaussian field, with
covariance structure corresponding to the \HH^{1/2}-norm of the test
functions. For this purpose, we obtain an approximate form of the explicit
formula, relying on Selberg's smoothed expression for and the
Helffer-Sj\"ostrand functional calculus. Our main result is an analogue of the
strong Szeg{\H o} theorem, known for Toeplitz operators and random matrix
theory
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