49 research outputs found
All the lowest order PDE for spectral gaps of Gaussian matrices
Tracy-Widom (TW) equations for one-matrix unitary ensembles (UE) (equivalent
to a particular case of Schlesinger equations for isomonodromic deformations)
are rewritten in a general form which allows one to derive all the lowest order
equations (PDE) for spectral gap probabilities of UE without intermediate
higher-order PDE. This is demonstrated on the example of Gaussian ensemble
(GUE) for which all the third order PDE for gap probabilities are obtained
explicitly. Moreover, there is a {\it second order} PDE for GUE probabilities
in the case of more than one spectral endpoint.
This approach allows to derive all PDE at once where possible, while in the
method based on Hirota bilinear identities and Virasoro constraints starting
with different bilinear identities leads to different subsets of the full set
of equations.Comment: 22 pages, references corrected, remark adde
Universal Structure and Universal PDE for Unitary Ensembles
An attempt is made to describe random matrix ensembles with unitary
invariance of measure (UE) in a unified way, using a combination of Tracy-Widom
(TW) and Adler-Shiota-Van Moerbeke (ASvM) approaches to derivation of partial
differential equations (PDE) for spectral gap probabilities. First, general
3-term recurrence relations for UE restricted to subsets of real line, or, in
other words, for functions in the resolvent kernel, are obtained. Using them,
simple universal relations between all TW dependent variables and
one-dimensional Toda lattice -functions are found. A universal system of
PDE for UE is derived from previous relations, which leads also to a {\it
single independent PDE} for spectral gap probability of various UE.
Thus, orthogonal function bases and Toda lattice are seen at the core of
correspondence of different approaches. Moreover, Toda-AKNS system provides a
common structure of PDE for unitary ensembles. Interestingly, this structure
can be seen in two very different forms: one arises from orthogonal
functions-Toda lattice considerations, while the other comes from Schlesinger
equations for isomonodromic deformations and their relation with TW equations.
The simple example of Gaussian matrices most neatly exposes this structure.Comment: 30 page
Beta ensembles, quantum Painlev\'e equations and isomonodromy systems
This is a review of recent developments in the theory of beta ensembles of
random matrices and their relations with conformal filed theory (CFT). There
are (almost) no new results here. This article can serve as a guide on
appearances and studies of quantum Painlev\'e and more general multidimensional
linear equations of Belavin-Polyakov-Zamolodchikov (BPZ) type in literature. We
demonstrate how BPZ equations of CFT arise from -ensemble eigenvalue
integrals. Quantum Painlev\'e equations are relatively simple instances of BPZ
or confluent BPZ equations, they are PDEs in two independent variables ("time"
and "space"). While CFT is known as quantum integrable theory, here we focus on
the appearing links of -ensembles and CFT with {\it classical}
integrable structure and isomonodromy systems. The central point is to show on
the example of quantum Painlev\'e II (QPII)~\cite{betaFP1} how classical
integrable structure can be extended to general values of (or CFT
central charge ), beyond the special cases () and
where its appearance is well-established. We also discuss an \'a
priori very different important approach, the ODE/IM correspondence giving
information about complex quantum integrable models, e.g.~CFT, from some
stationary Schr\"odinger ODEs. Solution of the ODEs depends on (discrete)
symmetries leading to functional equations for Stokes multipliers equivalent to
discrete integrable Hirota-type equations. The separation of "time" and "space"
variables, a consequence of our integrable structure, also leads to
Schr\"odinger ODEs and thus may have a connection with ODE/IM methods.Comment: Submitted to Cont. Mat
The correspondence between Tracy-Widom (TW) and Adler-Shiota-van Moerbeke (ASvM) approaches in random matrix theory: the Gaussian case
Two approaches (TW and ASvM) to derivation of integrable differential
equations for random matrix probabilities are compared. Both methods are
rewritten in such a form that simple and explicit relations between all TW
dependent variables and -functions of ASvM are found, for the example of
finite size Gaussian matrices. Orthogonal function systems and Toda lattice are
seen as the core structure of both approaches and their relationship.Comment: 20 pages, submitted to Journal of Mathematical Physic
A universal asymptotic regime in the hyperbolic nonlinear Schr\"odinger equation
The appearance of a fundamental long-time asymptotic regime in the two space
one time dimensional hyperbolic nonlinear Schr\"odinger (HNLS) equation is
discussed. Based on analytical and extensive numerical simulations an
approximate self-similar solution is found for a wide range of initial
conditions -- essentially for initial lumps of small to moderate energy. Even
relatively large initial amplitudes, which imply strong nonlinear effects,
eventually lead to local structures resembling those of the self-similar
solution, with appropriate small modifications. These modifications are
important in order to properly capture the behavior of the phase of the
solution. This solution has aspects that suggest it is a universal attractor
emanating from wide ranges of initial data.Comment: 36 pages, 26 pages text + 20 figure