11,319 research outputs found
Integrable Combinatorics
We review various combinatorial problems with underlying classical or quantum
integrable structures. (Plenary talk given at the International Congress of
Mathematical Physics, Aalborg, Denmark, August 10, 2012.)Comment: 21 pages, 16 figures, proceedings of ICMP1
Revisiting the combinatorics of the 2D Ising model
We provide a concise exposition with original proofs of combinatorial
formulas for the 2D Ising model partition function, multi-point fermionic
observables, spin and energy density correlations, for general graphs and
interaction constants, using the language of Kac-Ward matrices. We also give a
brief account of the relations between various alternative formalisms which
have been used in the combinatorial study of the planar Ising model: dimers and
Grassmann variables, spin and disorder operators, and, more recently,
s-holomorphic observables. In addition, we point out that these formulas can be
extended to the double-Ising model, defined as a pointwise product of two Ising
spin configurations on the same discrete domain, coupled along the boundary.Comment: Minor change in the notation (definition of eta). 55 pages, 4 figure
Statistical mechanics on isoradial graphs
Isoradial graphs are a natural generalization of regular graphs which give,
for many models of statistical mechanics, the right framework for studying
models at criticality. In this survey paper, we first explain how isoradial
graphs naturally arise in two approaches used by physicists: transfer matrices
and conformal field theory. This leads us to the fact that isoradial graphs
provide a natural setting for discrete complex analysis, to which we dedicate
one section. Then, we give an overview of explicit results obtained for
different models of statistical mechanics defined on such graphs: the critical
dimer model when the underlying graph is bipartite, the 2-dimensional critical
Ising model, random walk and spanning trees and the q-state Potts model.Comment: 22 page
Self-Assembly of Geometric Space from Random Graphs
We present a Euclidean quantum gravity model in which random graphs
dynamically self-assemble into discrete manifold structures. Concretely, we
consider a statistical model driven by a discretisation of the Euclidean
Einstein-Hilbert action; contrary to previous approaches based on simplicial
complexes and Regge calculus our discretisation is based on the Ollivier
curvature, a coarse analogue of the manifold Ricci curvature defined for
generic graphs. The Ollivier curvature is generally difficult to evaluate due
to its definition in terms of optimal transport theory, but we present a new
exact expression for the Ollivier curvature in a wide class of relevant graphs
purely in terms of the numbers of short cycles at an edge. This result should
be of independent intrinsic interest to network theorists. Action minimising
configurations prove to be cubic complexes up to defects; there are indications
that such defects are dynamically suppressed in the macroscopic limit. Closer
examination of a defect free model shows that certain classical configurations
have a geometric interpretation and discretely approximate vacuum solutions to
the Euclidean Einstein-Hilbert action. Working in a configuration space where
the geometric configurations are stable vacua of the theory, we obtain direct
numerical evidence for the existence of a continuous phase transition; this
makes the model a UV completion of Euclidean Einstein gravity. Notably, this
phase transition implies an area-law for the entropy of emerging geometric
space. Certain vacua of the theory can be interpreted as baby universes; we
find that these configurations appear as stable vacua in a mean field
approximation of our model, but are excluded dynamically whenever the action is
exact indicating the dynamical stability of geometric space. The model is
intended as a setting for subsequent studies of emergent time mechanisms.Comment: 26 pages, 9 figures, 2 appendice
The microscopic dynamics of quantum space as a group field theory
We provide a rather extended introduction to the group field theory approach
to quantum gravity, and the main ideas behind it. We present in some detail the
GFT quantization of 3d Riemannian gravity, and discuss briefly the current
status of the 4-dimensional extensions of this construction. We also briefly
report on recent results obtained in this approach and related open issues,
concerning both the mathematical definition of GFT models, and possible avenues
towards extracting interesting physics from them.Comment: 60 pages. Extensively revised version of the contribution to
"Foundations of Space and Time: Reflections on Quantum Gravity", edited by G.
Ellis, J. Murugan, A. Weltman, published by Cambridge University Pres
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