186 research outputs found
All order asymptotics of hyperbolic knot invariants from non-perturbative topological recursion of A-polynomials
Loop models on random maps via nested loops: case of domain symmetry breaking and application to the Potts model
We use the nested loop approach to investigate loop models on random planar
maps where the domains delimited by the loops are given two alternating colors,
which can be assigned different local weights, hence allowing for an explicit
Z_2 domain symmetry breaking. Each loop receives a non local weight n, as well
as a local bending energy which controls loop turns. By a standard cluster
construction that we review, the Q = n^2 Potts model on general random maps is
mapped to a particular instance of this problem with domain-non-symmetric
weights. We derive in full generality a set of coupled functional relations for
a pair of generating series which encode the enumeration of loop configurations
on maps with a boundary of a given color, and solve it by extending well-known
complex analytic techniques. In the case where loops are fully-packed, we
analyze in details the phase diagram of the model and derive exact equations
for the position of its non-generic critical points. In particular, we
underline that the critical Potts model on general random maps is not self-dual
whenever Q \neq 1. In a model with domain-symmetric weights, we also show the
possibility of a spontaneous domain symmetry breaking driven by the bending
energy.Comment: 44 pages, 13 figure
More on the O(n) model on random maps via nested loops: loops with bending energy
We continue our investigation of the nested loop approach to the O(n) model
on random maps, by extending it to the case where loops may visit faces of
arbitrary degree. This allows to express the partition function of the O(n)
loop model as a specialization of the multivariate generating function of maps
with controlled face degrees, where the face weights are determined by a fixed
point condition. We deduce a functional equation for the resolvent of the
model, involving some ring generating function describing the immediate
vicinity of the loops. When the ring generating function has a single pole, the
model is amenable to a full solution. Physically, such situation is realized
upon considering loops visiting triangles only and further weighting these
loops by some local bending energy. Our model interpolates between the two
previously solved cases of triangulations without bending energy and
quadrangulations with rigid loops. We analyze the phase diagram of our model in
details and derive in particular the location of its non-generic critical
points, which are in the universality classes of the dense and dilute O(n)
model coupled to 2D quantum gravity. Similar techniques are also used to solve
a twisting loop model on quadrangulations where loops are forced to make turns
within each visited square. Along the way, we revisit the problem of maps with
controlled, possibly unbounded, face degrees and give combinatorial derivations
of the one-cut lemma and of the functional equation for the resolvent.Comment: 40 pages, 9 figures, final accepted versio
Topological recursion and geometry
These are lecture notes for a 4h mini-course held in Toulouse, May 9-12th, at the thematic school on "Quantum topology and geometry". The goal of these lectures is to (a) explain some incarnations, in the last ten years, of the idea of topological recursion: in two dimensional quantum field theories, in cohomological field theories, in the computation of Weil-Petersson volumes of the moduli space of curves; (b) relate them more specifically to Eynard-Orantin topological recursion (revisited from Kontsevich-Soibelman point of view based on quantum Airy structures)
Asymptotic expansion of a partition function related to the sinh-model
This paper develops a method to carry out the large- asymptotic analysis
of a class of -dimensional integrals arising in the context of the so-called
quantum separation of variables method. We push further ideas developed in the
context of random matrices of size , but in the present problem, two scales
and naturally occur. In our case, the equilibrium measure
is -dependent and characterised by means of the solution to a
Riemann--Hilbert problem, whose large- behavior is analysed in
detail. Combining these results with techniques of concentration of measures
and an asymptotic analysis of the Schwinger-Dyson equations at the
distributional level, we obtain the large- behavior of the free energy
explicitly up to . The use of distributional Schwinger-Dyson is a novelty
that allows us treating sufficiently differentiable interactions and the mixing
of scales and , thus waiving the analyticity assumptions
often used in random matrix theory.Comment: 158 pages, 4 figures (V2 introduction extended, missprints corrected,
clarifications added to lemma 3.1.9 and corollary 3.1.10
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
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
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
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