227,022 research outputs found
Comments on worldsheet theories dual to free large N gauge theories
We continue to investigate properties of the worldsheet conformal field
theories (CFTs) which are conjectured to be dual to free large N gauge
theories, using the mapping of Feynman diagrams to the worldsheet suggested in
hep-th/0504229. The modular invariance of these CFTs is shown to be built into
the formalism. We show that correlation functions in these CFTs which are
localized on subspaces of the moduli space may be interpreted as delta-function
distributions, and that this can be consistent with a local worldsheet
description given some constraints on the operator product expansion
coefficients. We illustrate these features by a detailed analysis of a specific
four-point function diagram. To reliably compute this correlator we use a novel
perturbation scheme which involves an expansion in the large dimension of some
operators.Comment: 43 pages, 16 figures, JHEP format. v2: added reference
Critical percolation on certain non-unimodular graphs
An important conjecture in percolation theory is that almost surely no
infinite cluster exists in critical percolation on any transitive graph for
which the critical probability is less than 1. Earlier work has established
this for the amenable cases Z^2 and Z^d for large d, as well as for all
non-amenable graphs with unimodular automorphism groups. We show that the
conjecture holds for the basic classes of non-amenable graphs with
non-unimodular automorphism groups: for decorated trees and the non-unimodular
Diestel-Leader graphs. We also show that the connection probability between two
vertices decay exponentially in their distance. Finally, we prove that critical
percolation on the positive part of the lamplighter group has no infinite
clusters.Comment: 15 pages, 5 figures. Several corrections to previous versio
Reconciling long-term cultural diversity and short-term collective social behavior
An outstanding open problem is whether collective social phenomena occurring
over short timescales can systematically reduce cultural heterogeneity in the
long run, and whether offline and online human interactions contribute
differently to the process. Theoretical models suggest that short-term
collective behavior and long-term cultural diversity are mutually excluding,
since they require very different levels of social influence. The latter
jointly depends on two factors: the topology of the underlying social network
and the overlap between individuals in multidimensional cultural space.
However, while the empirical properties of social networks are well understood,
little is known about the large-scale organization of real societies in
cultural space, so that random input specifications are necessarily used in
models. Here we use a large dataset to perform a high-dimensional analysis of
the scientific beliefs of thousands of Europeans. We find that inter-opinion
correlations determine a nontrivial ultrametric hierarchy of individuals in
cultural space, a result unaccessible to one-dimensional analyses and in
striking contrast with random assumptions. When empirical data are used as
inputs in models, we find that ultrametricity has strong and counterintuitive
effects, especially in the extreme case of long-range online-like interactions
bypassing social ties. On short time-scales, it strongly facilitates a
symmetry-breaking phase transition triggering coordinated social behavior. On
long time-scales, it severely suppresses cultural convergence by restricting it
within disjoint groups. We therefore find that, remarkably, the empirical
distribution of individuals in cultural space appears to optimize the
coexistence of short-term collective behavior and long-term cultural diversity,
which can be realized simultaneously for the same moderate level of mutual
influence
Entropy in Dimension One
This paper completely classifies which numbers arise as the topological
entropy associated to postcritically finite self-maps of the unit interval.
Specifically, a positive real number h is the topological entropy of a
postcritically finite self-map of the unit interval if and only if exp(h) is an
algebraic integer that is at least as large as the absolute value of any of the
conjugates of exp(h); that is, if exp(h) is a weak Perron number. The
postcritically finite map may be chosen to be a polynomial all of whose
critical points are in the interval (0,1). This paper also proves that the weak
Perron numbers are precisely the numbers that arise as exp(h), where h is the
topological entropy associated to ergodic train track representatives of outer
automorphisms of a free group.Comment: 38 pages, 15 figures. This paper was completed by the author before
his death, and was uploaded by Dylan Thurston. A version including endnotes
by John Milnor will appear in the proceedings of the Banff conference on
Frontiers in Complex Dynamic
Ribbon Graphs, Quadratic Differentials on Riemann Surfaces, and Algebraic Curves Defined over
It is well known that there is a bijective correspondence between metric
ribbon graphs and compact Riemann surfaces with meromorphic Strebel
differentials. In this article, it is proved that Grothendieck's correspondence
between dessins d'enfants and Belyi morphisms is a special case of this
correspondence. For a metric ribbon graph with edge length 1, an algebraic
curve over and a Strebel differential on it is constructed. It is also
shown that the critical trajectories of the measured foliation that is
determined by the Strebel differential recover the original metric ribbon
graph. Conversely, for every Belyi morphism, a unique Strebel differential is
constructed such that the critical leaves of the measured foliation it
determines form a metric ribbon graph of edge length 1, which coincides with
the corresponding dessin d'enfant.Comment: Higher resolution figures available at
http://math.ucdavis.edu/~mulase
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