3,654 research outputs found
Martingale families and dimension in P
AbstractWe introduce a new measure notion on small complexity classes (called F-measure), based on martingale families, that gets rid of some drawbacks of previous measure notions: it can be used to define dimension because martingale families can make money on all strings, and it yields random sequences with an equal frequency of 0’s and 1’s. On larger complexity classes (E and above), F-measure is equivalent to Lutz resource-bounded measure. As applications to F-measure, we answer a question raised in [E. Allender, M. Strauss, Measure on small complexity classes, with application for BPP, in: Proc. of the 35th Ann. IEEE Symp. on Found. of Comp. Sci., 1994, pp. 807–818] by improving their result to: for almost every language A decidable in subexponential time, PA=BPPA. We show that almost all languages in PSPACE do not have small non-uniform complexity. We compare F-measure to previous notions and prove that martingale families are strictly stronger than Γ-measure [E. Allender, M. Strauss, Measure on small complexity classes, with application for BPP, in: Proc. of the 35th Ann. IEEE Symp. on Found. of Comp. Sci., 1994, pp. 807–818], we also discuss the limitations of martingale families concerning finite unions. We observe that all classes closed under polynomial many-one reductions have measure zero in EXP iff they have measure zero in SUBEXP. We use martingale families to introduce a natural generalization of Lutz resource-bounded dimension [J.H. Lutz, Dimension in complexity classes, in: Proceedings of the 15th Annual IEEE Conference on Computational Complexity, 2000, pp. 158–169] on P, which meets the intuition behind Lutz’s notion. We show that P-dimension lies between finite-state dimension and dimension on E. We prove an analogue of a Theorem of Eggleston in P, i.e. the class of languages whose characteristic sequence contains 1’s with frequency α, has dimension the Shannon entropy of α in P
Spatially independent martingales, intersections, and applications
We define a class of random measures, spatially independent martingales,
which we view as a natural generalisation of the canonical random discrete set,
and which includes as special cases many variants of fractal percolation and
Poissonian cut-outs. We pair the random measures with deterministic families of
parametrised measures , and show that under some natural
checkable conditions, a.s. the total measure of the intersections is H\"older
continuous as a function of . This continuity phenomenon turns out to
underpin a large amount of geometric information about these measures, allowing
us to unify and substantially generalize a large number of existing results on
the geometry of random Cantor sets and measures, as well as obtaining many new
ones. Among other things, for large classes of random fractals we establish (a)
very strong versions of the Marstrand-Mattila projection and slicing results,
as well as dimension conservation, (b) slicing results with respect to
algebraic curves and self-similar sets, (c) smoothness of convolutions of
measures, including self-convolutions, and nonempty interior for sumsets, (d)
rapid Fourier decay. Among other applications, we obtain an answer to a
question of I. {\L}aba in connection to the restriction problem for fractal
measures.Comment: 96 pages, 5 figures. v4: The definition of the metric changed in
Section 8. Polishing notation and other small changes. All main results
unchange
Hierarchical equilibria of branching populations
The objective of this paper is the study of the equilibrium behavior of a
population on the hierarchical group consisting of families of
individuals undergoing critical branching random walk and in addition these
families also develop according to a critical branching process. Strong
transience of the random walk guarantees existence of an equilibrium for this
two-level branching system. In the limit (called the hierarchical
mean field limit), the equilibrium aggregated populations in a nested sequence
of balls of hierarchical radius converge to a backward
Markov chain on . This limiting Markov chain can be explicitly
represented in terms of a cascade of subordinators which in turn makes possible
a description of the genealogy of the population.Comment: 62 page
Multiple Schramm-Loewner Evolutions and Statistical Mechanics Martingales
A statistical mechanics argument relating partition functions to martingales
is used to get a condition under which random geometric processes can describe
interfaces in 2d statistical mechanics at criticality. Requiring multiple SLEs
to satisfy this condition leads to some natural processes, which we study in
this note. We give examples of such multiple SLEs and discuss how a choice of
conformal block is related to geometric configuration of the interfaces and
what is the physical meaning of mixed conformal blocks. We illustrate the
general ideas on concrete computations, with applications to percolation and
the Ising model.Comment: 40 pages, 6 figures. V2: well, it looks better with the addresse
Sticky Particles and Stochastic Flows
Gaw\c{e}dzki and Horvai have studied a model for the motion of particles
carried in a turbulent fluid and shown that in a limiting regime with low
levels of viscosity and molecular diffusivity, pairs of particles exhibit the
phenomena of stickiness when they meet. In this paper we characterise the
motion of an arbitrary number of particles in a simplified version of their
model
Towards conformal invariance of 2D lattice models
Many 2D lattice models of physical phenomena are conjectured to have
conformally invariant scaling limits: percolation, Ising model, self-avoiding
polymers, ... This has led to numerous exact (but non-rigorous) predictions of
their scaling exponents and dimensions. We will discuss how to prove the
conformal invariance conjectures, especially in relation to Schramm-Loewner
Evolution.Comment: ICM 2006 paper with a few typos correcte
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