26,738 research outputs found
Mean field spin glasses treated with PDE techniques
Following an original idea of F. Guerra, in this notes we analyze the
Sherrington-Kirkpatrick model from different perspectives, all sharing the
underlying approach which consists in linking the resolution of the statistical
mechanics of the model (e.g. solving for the free energy) to well-known partial
differential equation (PDE) problems (in suitable spaces). The plan is then to
solve the related PDE using techniques involved in their native field and
lastly bringing back the solution in the proper statistical mechanics
framework. Within this strand, after a streamlined test-case on the Curie-Weiss
model to highlight the methods more than the physics behind, we solve the SK
both at the replica symmetric and at the 1-RSB level, obtaining the correct
expression for the free energy via an analogy to a Fourier equation and for the
self-consistencies with an analogy to a Burger equation, whose shock wave
develops exactly at critical noise level (triggering the phase transition). Our
approach, beyond acting as a new alternative method (with respect to the
standard routes) for tackling the complexity of spin glasses, links symmetries
in PDE theory with constraints in statistical mechanics and, as a novel result
from the theoretical physics perspective, we obtain a new class of polynomial
identities (namely of Aizenman-Contucci type but merged within the Guerra's
broken replica measures), whose interest lies in understanding, via the recent
Panchenko breakthroughs, how to force the overlap organization to the
ultrametric tree predicted by Parisi
Models of q-algebra representations: q-integral transforms and "addition theorems''
In his classic book on group representations and special functions Vilenkin studied the matrix elements of irreducible representations of the Euclidean and oscillator Lie algebras with respect to countable bases of eigenfunctions of the Cartan subalgebras, and he computed the summation identities for Bessel functions and Laguerre polynomials associated with the addition theorems for these matrix elements. He also studied matrix elements of the pseudo-Euclidean and pseudo-oscillator algebras with respect to the continuum bases of generalized eigenfunctions of the Cartan subalgebras of these Lie algebras and this resulted in realizations of the addition theorems for the matrix elements as integral transform identities for Bessel functions and for confluent hypergeometric functions. Here we work out q analogs of these results in which the usual exponential function mapping from the Lie algebra to the Lie group is replaced by the q-exponential mappings Eq and eq. This study of representations of the Euclidean quantum algebra and the q-oscillator algebra (not a quantum algebra) leads to summation, integral transform, and q-integral transform identities for q analogs of the Bessel and confluent hypergeometric functions, extending the results of Vilenkin for the q=1 case
The Cosmological Bootstrap: Spinning Correlators from Symmetries and Factorization
We extend the cosmological bootstrap to correlators involving massless
particles with spin. In de Sitter space, these correlators are constrained both
by symmetries and by locality. In particular, the de Sitter isometries become
conformal symmetries on the future boundary of the spacetime, which are
reflected in a set of Ward identities that the boundary correlators must
satisfy. We solve these Ward identities by acting with weight-shifting
operators on scalar seed solutions. Using this weight-shifting approach, we
derive three- and four-point correlators of massless spin-1 and spin-2 fields
with conformally coupled scalars. Four-point functions arising from tree-level
exchange are singular in particular kinematic configurations, and the
coefficients of these singularities satisfy certain factorization properties.
We show that in many cases these factorization limits fix the structure of the
correlators uniquely, without having to solve the conformal Ward identities.
The additional constraint of locality for massless spinning particles manifests
itself as current conservation on the boundary. We find that the four-point
functions only satisfy current conservation if the s, t, and u-channels are
related to each other, leading to nontrivial constraints on the couplings
between the conserved currents and other operators in the theory. For spin-1
currents this implies charge conservation, while for spin-2 currents we recover
the equivalence principle from a purely boundary perspective. For multiple
spin-1 fields, we recover the structure of Yang-Mills theory. Finally, we apply
our methods to slow-roll inflation and derive a few phenomenologically relevant
scalar-tensor three-point functions.Comment: 128 pages, 15 figures; V3: minor corrections and references adde
Multidimensional Matrix Inversions and Elliptic Hypergeometric Series on Root Systems
Multidimensional matrix inversions provide a powerful tool for studying
multiple hypergeometric series. In order to extend this technique to elliptic
hypergeometric series, we present three new multidimensional matrix inversions.
As applications, we obtain a new elliptic Jackson summation, as well as
several quadratic, cubic and quartic summation formulas
- …