15,335 research outputs found
Energy landscapes, lowest gaps, and susceptibility of elastic manifolds at zero temperature
We study the effect of an external field on (1+1) and (2+1) dimensional
elastic manifolds, at zero temperature and with random bond disorder. Due to
the glassy energy landscape the configuration of a manifold changes often in
abrupt, ``first order'' -type of large jumps when the field is applied. First
the scaling behavior of the energy gap between the global energy minimum and
the next lowest minimum of the manifold is considered, by employing exact
ground state calculations and an extreme statistics argument. The scaling has a
logarithmic prefactor originating from the number of the minima in the
landscape, and reads ,
where is the roughness exponent and is the energy fluctuation
exponent of the manifold, is the linear size of the manifold, and is
the system height. The gap scaling is extended to the case of a finite external
field and yields for the susceptibility of the manifolds . We also present a mean field argument
for the finite size scaling of the first jump field, .
The implications to wetting in random systems, to finite-temperature behavior
and the relation to Kardar-Parisi-Zhang non-equilibrium surface growth are
discussed.Comment: 20 pages, 22 figures, accepted for publication in Eur. Phys. J.
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
Extremal statistics in the energetics of domain walls
We study at T=0 the minimum energy of a domain wall and its gap to the first
excited state concentrating on two-dimensional random-bond Ising magnets. The
average gap scales as , where , is the energy fluctuation exponent, length scale, and
the number of energy valleys. The logarithmic scaling is due to extremal
statistics, which is illustrated by mapping the problem into the
Kardar-Parisi-Zhang roughening process. It follows that the susceptibility of
domain walls has also a logarithmic dependence on system size.Comment: Accepted for publication in Phys. Rev.
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