18,801 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
Mixing rate in infinite measure for Z^d-extension, application to the periodic Sinai billiard
We study the rate of mixing of observables of Z^d-extensions of probability
preserving dynamical systems. We explain how this question is directly linked
to the local limit theorem and establish a rate of mixing for general classes
of observables of the Z^2-periodic Sinai billiard. We compare our approach with
the induction method
On-the-fly Uniformization of Time-Inhomogeneous Infinite Markov Population Models
This paper presents an on-the-fly uniformization technique for the analysis
of time-inhomogeneous Markov population models. This technique is applicable to
models with infinite state spaces and unbounded rates, which are, for instance,
encountered in the realm of biochemical reaction networks. To deal with the
infinite state space, we dynamically maintain a finite subset of the states
where most of the probability mass is located. This approach yields an
underapproximation of the original, infinite system. We present experimental
results to show the applicability of our technique
Quantum noise, entanglement and chaos in the quantum field theory of mind/brain states
We review the dissipative quantum model of brain and present recent
developments related with the r\^ole of entanglement, quantum noise and chaos
in the model.Comment: 15 page
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