148 research outputs found
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
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
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
Special cases of the orbifold version of Zvonkine's -ELSV formula
We prove the orbifold version of Zvonkine's -ELSV formula in two special cases: the case of (complete -cycles) for any genus and the case of any for genus
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
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