27,453 research outputs found
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
A counterexample to Thiagarajan's conjecture on regular event structures
We provide a counterexample to a conjecture by Thiagarajan (1996 and 2002)
that regular event structures correspond exactly to event structures obtained
as unfoldings of finite 1-safe Petri nets. The same counterexample is used to
disprove a closely related conjecture by Badouel, Darondeau, and Raoult (1999)
that domains of regular event structures with bounded -cliques are
recognizable by finite trace automata. Event structures, trace automata, and
Petri nets are fundamental models in concurrency theory. There exist nice
interpretations of these structures as combinatorial and geometric objects.
Namely, from a graph theoretical point of view, the domains of prime event
structures correspond exactly to median graphs; from a geometric point of view,
these domains are in bijection with CAT(0) cube complexes.
A necessary condition for both conjectures to be true is that domains of
regular event structures (with bounded -cliques) admit a regular nice
labeling. To disprove these conjectures, we describe a regular event domain
(with bounded -cliques) that does not admit a regular nice labeling.
Our counterexample is derived from an example by Wise (1996 and 2007) of a
nonpositively curved square complex whose universal cover is a CAT(0) square
complex containing a particular plane with an aperiodic tiling. We prove that
other counterexamples to Thiagarajan's conjecture arise from aperiodic 4-way
deterministic tile sets of Kari and Papasoglu (1999) and Lukkarila (2009).
On the positive side, using breakthrough results by Agol (2013) and Haglund
and Wise (2008, 2012) from geometric group theory, we prove that Thiagarajan's
conjecture is true for regular event structures whose domains occur as
principal filters of hyperbolic CAT(0) cube complexes which are universal
covers of finite nonpositively curved cube complexes
Low energy analysis of pi N --> pi N
We derive a representation for the pion nucleon scattering amplitude that is
valid to the fourth order of the chiral expansion. To obtain the correct
analytic structure of the singularities in the low energy region, we have
performed the calculation in a relativistic framework (infrared
regularization). The result can be written in terms of functions of a single
variable. We study the corresponding dispersion relations and discuss the
problems encountered in the straightforward nonrelativistic expansion of the
infrared singularities. As an application, we evaluate the corrections to the
Goldberger-Treiman relation and to the low energy theorem that relates the
value of the amplitude at the Cheng-Dashen point to the \sigma-term. While
chiral symmetry does govern the behaviour of the amplitude in the vicinity of
this point, the representation for the scattering amplitude is not accurate
enough to use it for an extrapolation of the experimental data to the
subthreshold region. We propose to perform this extrapolation on the basis of a
set of integral equations that interrelate the lowest partial waves and are
analogous to the Roy equations for \pi\pi scattering.Comment: 97 pages (LaTeX), 16 figures. Two references added, correction in
table one. Published versio
Kertesz on Fat Graphs?
The identification of phase transition points, beta_c, with the percolation
thresholds of suitably defined clusters of spins has proved immensely fruitful
in many areas of statistical mechanics. Some time ago Kertesz suggested that
such percolation thresholds for models defined in field might also have
measurable physical consequences for regions of the phase diagram below beta_c,
giving rise to a ``Kertesz line'' running between beta_c and the bond
percolation threshold, beta_p, in the M, beta plane.
Although no thermodynamic singularities were associated with this line it
could still be divined by looking for a change in the behaviour of high-field
series for quantities such as the free energy or magnetisation. Adler and
Stauffer did precisely this with some pre-existing series for the regular
square lattice and simple cubic lattice Ising models and did, indeed, find
evidence for such a change in high-field series around beta_p. Since there is a
general dearth of high-field series there has been no other work along these
lines.
In this paper we use the solution of the Ising model in field on planar
random graphs by Boulatov and Kazakov to carry out a similar exercise for the
Ising model on random graphs (i.e. coupled to 2D quantum gravity). We generate
a high-field series for the Ising model on random graphs and examine
its behaviour for evidence of a Kertesz line
Social Dilemmas and Cooperation in Complex Networks
In this paper we extend the investigation of cooperation in some classical
evolutionary games on populations were the network of interactions among
individuals is of the scale-free type. We show that the update rule, the payoff
computation and, to some extent the timing of the operations, have a marked
influence on the transient dynamics and on the amount of cooperation that can
be established at equilibrium. We also study the dynamical behavior of the
populations and their evolutionary stability.Comment: 12 pages, 7 figures. to appea
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