106 research outputs found
Statistical Mechanics of Self-Avoiding Manifolds (Part II)
We consider a model of a D-dimensional tethered manifold interacting by
excluded volume in R^d with a single point. Use of intrinsic distance geometry
provides a rigorous definition of the analytic continuation of the perturbative
expansion for arbitrary D, 0 < D < 2. Its one-loop renormalizability is first
established by direct resummation. A renormalization operation R is then
described, which ensures renormalizability to all orders. The similar question
of the renormalizability of the self-avoiding manifold (SAM) Edwards model is
then considered, first at one-loop, then to all orders. We describe a
short-distance multi-local operator product expansion, which extends methods of
local field theories to a large class of models with non-local singular
interactions. It vindicates the direct renormalization method used earlier in
part I of these lectures, as well as the corresponding scaling laws.Comment: 32 pages, 9 figures, Second Part and extensive update of Lecture
Notes originally given in ``Statistical Mechanics of Membranes and
Surfaces'', Fifth Jerusalem Winter School for Theoretical Physics (1987), D.
R. Nelson, T. Piran,and S. Weinberg ed
Geometry of the Casimir Effect
When the vacuum is partitioned by material boundaries with arbitrary shape,
one can define the zero-point energy and the free energy of the electromagnetic
waves in it: this can be done, independently of the nature of the boundaries,
in the limit that they become perfect conductors, provided their curvature is
finite. The first examples we consider are Casimir's original configuration of
parallel plates, and the experimental situation of a sphere in front of a
plate. For arbitrary geometries, we give an explicit expression for the
zero-point energy and the free energy in terms of an integral kernel acting on
the boundaries; it can be expanded in a convergent series interpreted as a
succession of an even number of scatterings of a wave. The quantum and thermal
fluctuations of vacuum then appear as a purely geometric property. The Casimir
effect thus defined exists only owing to the electromagnetic nature of the
field. It does not exist for thin foils with sharp folds, but Casimir forces
between solid wedges are finite. We work out various applications: low
temperature, high temperature where wrinkling constraints appear, stability of
a plane foil, transfer of energy from one side of a curved boundary to the
other, forces between distant conductors, special shapes (parallel plates,
sphere, cylinder, honeycomb).Comment: 44 pages, 8 figures; Proceedings of the 15 th SIGRAV Conference on
General Relativity and Gravitational Physics, Villa Mondragone, Monte Porzio
Catone, Roma, Italy, September 9-12, 200
Liouville Quantum Gravity and KPZ
Consider a bounded planar domain D, an instance h of the Gaussian free field
on D (with Dirichlet energy normalized by 1/(2\pi)), and a constant 0 < gamma <
2. The Liouville quantum gravity measure on D is the weak limit as epsilon
tends to 0 of the measures \epsilon^{\gamma^2/2} e^{\gamma h_\epsilon(z)}dz,
where dz is Lebesgue measure on D and h_\epsilon(z) denotes the mean value of h
on the circle of radius epsilon centered at z. Given a random (or
deterministic) subset X of D one can define the scaling dimension of X using
either Lebesgue measure or this random measure. We derive a general quadratic
relation between these two dimensions, which we view as a probabilistic
formulation of the KPZ relation from conformal field theory. We also present a
boundary analog of KPZ (for subsets of the boundary of D). We discuss the
connection between discrete and continuum quantum gravity and provide a
framework for understanding Euclidean scaling exponents via quantum gravity.Comment: 56 pages. Revised version contains more details. To appear in
Inventione
Nesting statistics in the loop model on random planar maps
In the loop model on random planar maps, we study the depth -- in
terms of the number of levels of nesting -- of the loop configuration, by means
of analytic combinatorics. We focus on the `refined' generating series of
pointed disks or cylinders, which keep track of the number of loops separating
the marked point from the boundary (for disks), or the two boundaries (for
cylinders). For the general loop model, we show that these generating
series satisfy functional relations obtained by a modification of those
satisfied by the unrefined generating series. In a more specific model
where loops cross only triangles and have a bending energy, we explicitly
compute the refined generating series. We analyze their non generic critical
behavior in the dense and dilute phases, and obtain the large deviations
function of the nesting distribution, which is expected to be universal. Using
the framework of Liouville quantum gravity (LQG), we show that a rigorous
functional KPZ relation can be applied to the multifractal spectrum of extreme
nesting in the conformal loop ensemble () in the Euclidean
unit disk, as obtained by Miller, Watson and Wilson, or to its natural
generalization to the Riemann sphere. It allows us to recover the large
deviations results obtained for the critical random planar map models.
This offers, at the refined level of large deviations theory, a rigorous check
of the fundamental fact that the universal scaling limits of random planar map
models as weighted by partition functions of critical statistical models are
given by LQG random surfaces decorated by independent CLEs.Comment: 71 pages, 11 figures. v2: minor text and abstract edits, references
adde
A Bijection between well-labelled positive paths and matchings
A well-labelled positive path of size n is a pair (p,\sigma) made of a word
p=p_1p_2...p_{n-1} on the alphabet {-1, 0,+1} such that the sum of the letters
of any prefix is non-negative, together with a permutation \sigma of
{1,2,...,n} such that p_i=-1 implies \sigma(i)<\sigma(i+1), while p_i=1 implies
\sigma(i)>\sigma(i+1). We establish a bijection between well-labelled positive
paths of size and matchings (i.e. fixed-point free involutions) on
{1,2,...,2n}. This proves that the number of well-labelled positive paths is
(2n-1)!!. By specialising our bijection, we also prove that the number of
permutations of size n such that each prefix has no more ascents than descents
is [(n-1)!!]^2 if n is even and n!!(n-2)!! otherwise. Our result also prove
combinatorially that the n-dimensional polytope consisting of all points
(x_1,...,x_n) in [-1,1]^n such that the sum of the first j coordinates is
non-negative for all j=1,2,...,n has volume (2n-1)!!/n!
Integral means spectrum of whole-plane SLE
We complete the mathematical analysis of the fine structure of harmonic
measure on SLE curves that was initiated by Beliaev and Smirnov, as described
by the averaged integral means spectrum. For the unbounded version of
whole-plane SLE as studied by Duplantier, Nguyen, Nguyen and Zinsmeister, and
Loutsenko and Yermolayeva, a phase transition has been shown to occur for high
enough moments from the bulk spectrum towards a novel spectrum related to the
point at infinity. For the bounded version of whole-plane SLE studied here, a
similar transition phenomenon, now associated with the SLE origin, is proved to
exist for low enough moments, but we show that it is superseded by the earlier
occurrence of the transition to the SLE tip spectrum.Comment: 14 pages, 1 figure; final versio
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