6,281 research outputs found
Concavity analysis of the tangent method
The tangent method has recently been devised by Colomo and Sportiello
(arXiv:1605.01388 [math-ph]) as an efficient way to determine the shape of
arctic curves. Largely conjectural, it has been tested successfully in a
variety of models. However no proof and no general geometric insight have been
given so far, either to show its validity or to allow for an understanding of
why the method actually works. In this paper, we propose a universal framework
which accounts for the tangency part of the tangent method, whenever a
formulation in terms of directed lattice paths is available. Our analysis shows
that the key factor responsible for the tangency property is the concavity of
the entropy (also called the Lagrangean function) of long random lattice paths.
We extend the proof of the tangency to -deformed paths.Comment: published version, 22 page
Probability around the Quantum Gravity. Part 1: Pure Planar Gravity
In this paper we study stochastic dynamics which leaves quantum gravity
equilibrium distribution invariant. We start theoretical study of this dynamics
(earlier it was only used for Monte-Carlo simulation). Main new results concern
the existence and properties of local correlation functions in the
thermodynamic limit. The study of dynamics constitutes a third part of the
series of papers where more general class of processes were studied (but it is
self-contained), those processes have some universal significance in
probability and they cover most concrete processes, also they have many
examples in computer science and biology. At the same time the paper can serve
an introduction to quantum gravity for a probabilist: we give a rigorous
exposition of quantum gravity in the planar pure gravity case. Mostly we use
combinatorial techniques, instead of more popular in physics random matrix
models, the central point is the famous exponent.Comment: 40 pages, 11 figure
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
Machine Hyperconsciousness
Individual animal consciousness appears limited to a single giant component of interacting cognitive modules, instantiating a shifting, highly tunable, Global Workspace. Human institutions, by contrast, can support several, often many, such giant components simultaneously, although they generally function far more slowly than the minds of the individuals who compose them. Machines having multiple global workspaces -- hyperconscious machines -- should, however, be able to operate at the few hundred milliseconds characteistic of individual consciousness. Such multitasking -- machine or institutional -- while clearly limiting the phenomenon of inattentional blindness, does not eliminate it, and introduces characteristic failure modes involving the distortion of information sent between global workspaces. This suggests that machines explicitly designed along these principles, while highly efficient at certain sets of tasks, remain subject to canonical and idiosyncratic failure patterns analogous to, but more complicated than, those explored in Wallace (2006a). By contrast, institutions, facing similar challenges, are usually deeply embedded in a highly stabilizing cultural matrix of law, custom, and tradition which has evolved over many centuries. Parallel development of analogous engineering strategies, directed toward ensuring an 'ethical' device, would seem requisite to the sucessful application of any form of hyperconscious machine technology
The critical probability for random Voronoi percolation in the plane is 1/2
We study percolation in the following random environment: let be a
Poisson process of constant intensity in the plane, and form the Voronoi
tessellation of the plane with respect to . Colour each Voronoi cell black
with probability , independently of the other cells. We show that the
critical probability is 1/2. More precisely, if then the union of the
black cells contains an infinite component with probability 1, while if
then the distribution of the size of the component of black cells containing a
given point decays exponentially. These results are analogous to Kesten's
results for bond percolation in the square lattice.
The result corresponding to Harris' Theorem for bond percolation in the
square lattice is known: Zvavitch noted that one of the many proofs of this
result can easily be adapted to the random Voronoi setting. For Kesten's
results, none of the existing proofs seems to adapt. The methods used here also
give a new and very simple proof of Kesten's Theorem for the square lattice; we
hope they will be applicable in other contexts as well.Comment: 55 pages, 20 figures; minor changes; to appear in Probability Theory
and Related Field
Minimal stretch maps between hyperbolic surfaces
This paper develops a theory of Lipschitz comparisons of hyperbolic surfaces
analogous to the theory of quasi-conformal comparisons. Extremal Lipschitz maps
(minimal stretch maps) and geodesics for the `Lipschitz metric' are
constructed. The extremal Lipschitz constant equals the maximum ratio of
lengths of measured laminations, which is attained with probability one on a
simple closed curve. Cataclysms are introduced, generalizing earthquakes by
permitting more violent shearing in both directions along a fault. Cataclysms
provide useful coordinates for Teichmuller space that are convenient for
computing derivatives of geometric function in Teichmuller space and measured
lamination space.Comment: 53 pages, 11 figures, version of 1986 preprin
- âŠ