134 research outputs found
Higher Order Analogues of Tracy-Widom Distributions via the Lax Method
We study the distribution of the largest eigenvalue in formal Hermitian
one-matrix models at multicriticality, where the spectral density acquires an
extra number of k-1 zeros at the edge. The distributions are directly expressed
through the norms of orthogonal polynomials on a semi-infinite interval, as an
alternative to using Fredholm determinants. They satisfy non-linear recurrence
relations which we show form a Lax pair, making contact to the string
literature in the early 1990's. The technique of pseudo-differential operators
allows us to give compact expressions for the logarithm of the gap probability
in terms of the Painleve XXXIV hierarchy. These are the higher order analogues
of the Tracy-Widom distribution which has k=1. Using known Backlund
transformations we show how to simplify earlier equivalent results that are
derived from Fredholm determinant theory, valid for even k in terms of the
Painleve II hierarchy.Comment: 24 pages. Improved discussion of Backlund transformations, in
addition to other minor improvements in text. Typos corrected. Matches
published versio
Large deviations of the maximal eigenvalue of random matrices
We present detailed computations of the 'at least finite' terms (three
dominant orders) of the free energy in a one-cut matrix model with a hard edge
a, in beta-ensembles, with any polynomial potential. beta is a positive number,
so not restricted to the standard values beta = 1 (hermitian matrices), beta =
1/2 (symmetric matrices), beta = 2 (quaternionic self-dual matrices). This
model allows to study the statistic of the maximum eigenvalue of random
matrices. We compute the large deviation function to the left of the expected
maximum. We specialize our results to the gaussian beta-ensembles and check
them numerically. Our method is based on general results and procedures already
developed in the literature to solve the Pastur equations (also called "loop
equations"). It allows to compute the left tail of the analog of Tracy-Widom
laws for any beta, including the constant term.Comment: 62 pages, 4 figures, pdflatex ; v2 bibliography corrected ; v3 typos
corrected and preprint added ; v4 few more numbers adde
Purity distribution for generalized random Bures mixed states
We compute the distribution of the purity for random density matrices
(i.e.random mixed states) in a large quantum system, distributed according to
the Bures measure. The full distribution of the purity is computed using a
mapping to random matrix theory and then a Coulomb gas method. We find three
regimes that correspond to two phase transitions in the associated Coulomb gas.
The first transition is characterized by an explosion of the third derivative
on the left of the transition point. The second transition is of first order,
it is characterized by the detachement of a single charge of the Coulomb gas. A
key remark in this paper is that the random Bures states are closely related to
the O(n) model for n=1. This actually led us to study "generalized Bures
states" by keeping general instead of specializing to n=1
A recursive approach to the O(n) model on random maps via nested loops
We consider the O(n) loop model on tetravalent maps and show how to rephrase
it into a model of bipartite maps without loops. This follows from a
combinatorial decomposition that consists in cutting the O(n) model
configurations along their loops so that each elementary piece is a map that
may have arbitrary even face degrees. In the induced statistics, these maps are
drawn according to a Boltzmann distribution whose parameters (the face weights)
are determined by a fixed point condition. In particular, we show that the
dense and dilute critical points of the O(n) model correspond to bipartite maps
with large faces (i.e. whose degree distribution has a fat tail). The
re-expression of the fixed point condition in terms of linear integral
equations allows us to explore the phase diagram of the model. In particular,
we determine this phase diagram exactly for the simplest version of the model
where the loops are "rigid". Several generalizations of the model are
discussed.Comment: 47 pages, 13 figures, final version (minor changes with v2 after
proof corrections
Right tail expansion of Tracy-Widom beta laws
Using loop equations, we compute the large deviation function of the maximum
eigenvalue to the right of the spectrum in the Gaussian beta matrix ensembles,
to all orders in 1/N. We then give a physical derivation of the all order
asymptotic expansion of the right tail Tracy-Widom beta laws, for all positive
beta, by studying the double scaling limit.Comment: 23 page
Brunel-Dominated Proton Acceleration with a Few-Cycle Laser Pulse
International audienceExperimental measurements of backward accelerated protons are presented. The beam is produced when an ultrashort (5 fs) laser pulse, delivered by a kHz laser system, with a high temporal contrast (10 8), interacts with a thick solid target. Under these conditions, proton cutoff energy dependence with laser parameters, such as pulse energy, polarization (from p to s), and pulse duration (from 5 to 500 fs), is studied. Theoretical model and two-dimensional particle-in-cell simulations, in good agreement with a large set of experimental results, indicate that proton acceleration is directly driven by Brunel electrons, in contrast to conventional target normal sheath acceleration that relies on electron thermal pressure
Spectral density asymptotics for Gaussian and Laguerre -ensembles in the exponentially small region
The first two terms in the large asymptotic expansion of the
moment of the characteristic polynomial for the Gaussian and Laguerre
-ensembles are calculated. This is used to compute the asymptotic
expansion of the spectral density in these ensembles, in the exponentially
small region outside the leading support, up to terms . The leading form
of the right tail of the distribution of the largest eigenvalue is given by the
density in this regime. It is demonstrated that there is a scaling from this,
to the right tail asymptotics for the distribution of the largest eigenvalue at
the soft edge.Comment: 19 page
Resolvent methods for steady premixed flame shapes governed by the Zhdanov-Trubnikov equation
Using pole decompositions as starting points, the one parameter (-1 =< c < 1)
nonlocal and nonlinear Zhdanov-Trubnikov (ZT) equation for the steady shapes of
premixed gaseous flames is studied in the large-wrinkle limit. The singular
integral equations for pole densities are closely related to those satisfied by
the spectral density in the O(n) matrix model, with n = -2(1 + c)/(1 - c). They
can be solved via the introduction of complex resolvents and the use of complex
analysis. We retrieve results obtained recently for -1 =< c =< 0, and we
explain and cure their pathologies when they are continued naively to 0 < c <
1. Moreover, for any -1 =< c < 1, we derive closed-form expressions for the
shapes of steady isolated flame crests, and then bicoalesced periodic fronts.
These theoretical results fully agree with numerical resolutions. Open problems
are evoked.Comment: v2: 29 pages, 6 figures, some typos correcte
Enumeration of maps with self avoiding loops and the O(n) model on random lattices of all topologies
We compute the generating functions of a O(n) model (loop gas model) on a
random lattice of any topology. On the disc and the cylinder, they were already
known, and here we compute all the other topologies. We find that the
generating functions (and the correlation functions of the lattice) obey the
topological recursion, as usual in matrix models, i.e they are given by the
symplectic invariants of their spectral curve.Comment: pdflatex, 89 pages, 12 labelled figures (15 figures at all), minor
correction
A unified fluctuation formula for one-cut -ensembles of random matrices
Using a Coulomb gas approach, we compute the generating function of the
covariances of power traces for one-cut -ensembles of random matrices in
the limit of large matrix size. This formula depends only on the support of the
spectral density, and is therefore universal for a large class of models. This
allows us to derive a closed-form expression for the limiting covariances of an
arbitrary one-cut -ensemble. As particular cases of the main result we
consider the classical -Gaussian, -Wishart and -Jacobi
ensembles, for which we derive previously available results as well as new ones
within a unified simple framework. We also discuss the connections between the
problem of trace fluctuations for the Gaussian Unitary Ensemble and the
enumeration of planar maps.Comment: 16 pages, 4 figures, 3 tables. Revised version where references have
been added and typos correcte
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