73 research outputs found
Feller property and infinitesimal generator of the exploration process
We consider the exploration process associated to the continuous random tree
(CRT) built using a Levy process with no negative jumps. This process has been
studied by Duquesne, Le Gall and Le Jan. This measure-valued Markov process is
a useful tool to study CRT as well as super-Brownian motion with general
branching mechanism. In this paper we prove this process is Feller, and we
compute its infinitesimal generator on exponential functionals and give the
corresponding martingale
The topological structure of scaling limits of large planar maps
We discuss scaling limits of large bipartite planar maps. If p is a fixed
integer strictly greater than 1, we consider a random planar map M(n) which is
uniformly distributed over the set of all 2p-angulations with n faces. Then, at
least along a suitable subsequence, the metric space M(n) equipped with the
graph distance rescaled by the factor n to the power -1/4 converges in
distribution as n tends to infinity towards a limiting random compact metric
space, in the sense of the Gromov-Hausdorff distance. We prove that the
topology of the limiting space is uniquely determined independently of p, and
that this space can be obtained as the quotient of the Continuum Random Tree
for an equivalence relation which is defined from Brownian labels attached to
the vertices. We also verify that the Hausdorff dimension of the limit is
almost surely equal to 4.Comment: 45 pages Second version with minor modification
Exponential Separation of Quantum and Classical Online Space Complexity
Although quantum algorithms realizing an exponential time speed-up over the
best known classical algorithms exist, no quantum algorithm is known performing
computation using less space resources than classical algorithms. In this
paper, we study, for the first time explicitly, space-bounded quantum
algorithms for computational problems where the input is given not as a whole,
but bit by bit. We show that there exist such problems that a quantum computer
can solve using exponentially less work space than a classical computer. More
precisely, we introduce a very natural and simple model of a space-bounded
quantum online machine and prove an exponential separation of classical and
quantum online space complexity, in the bounded-error setting and for a total
language. The language we consider is inspired by a communication problem (the
set intersection function) that Buhrman, Cleve and Wigderson used to show an
almost quadratic separation of quantum and classical bounded-error
communication complexity. We prove that, in the framework of online space
complexity, the separation becomes exponential.Comment: 13 pages. v3: minor change
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