338 research outputs found
Quenched LDP for words in a letter sequence
When we cut an i.i.d. sequence of letters into words according to an independent renewal process, we obtain an i.i.d. sequence of words. In the annealed large deviation principle (LDP) for the empirical process of words, the rate function is the specific relative entropy of the observed law of words w.r.t. the reference law of words. In the present paper we consider the quenched LDP, i.e., we condition on a typical letter sequence. We focus on the case where the renewal process has an algebraic tail. The rate function turns out to be a sum of two terms, one being the annealed rate function, the other being proportional to the specific relative entropy of the observed law of letters w.r.t. the reference law of letters, with the former being obtained by concatenating the words and randomising the location of the origin. The proportionality constant equals the tail exponent of the renewal process. Earlier work by Birkner considered the case where the renewal process has an exponential tail, in which case the rate function turns out to be the first term on the set where the second term vanishes and to be infinite elsewhere. We apply our LDP to prove that the radius of convergence of the moment generating function of the collision local time of two strongly transient random walks on Zd, d = 1, strictly increases when we condition on one of the random walks, both in discrete time and in continuous time. The presence of these gaps implies the existence of an intermediate phase for the long-time behaviour of a class of coupled branching processes, interacting diffusions, respectively, directed polymers in random environments
Collision local time of transient random walks and intermediate phases in interacting stochastic systems
In a companion paper, a quenched large deviation principle (LDP) has been established for the empirical process of words obtained by cutting an i.i.d. sequence of letters into words according to a renewal process. We apply this LDP to prove that the radius of convergence of the moment generating function of the collision local time of two strongly transient random walks on Zd, d = 1, strictly increases when we condition on one of the random walks, both in discrete time and in continuous time. We conjecture that the same holds for two transient but not strongly transient random walks. The presence of these gaps implies the existence of an intermediate phase for the long-time behaviour of a class of coupled branching processes, interacting diffusions, respectively, directed polymers in random environments
On Exceptional Times for generalized Fleming-Viot Processes with Mutations
If is a standard Fleming-Viot process with constant mutation rate
(in the infinitely many sites model) then it is well known that for each
the measure is purely atomic with infinitely many atoms. However,
Schmuland proved that there is a critical value for the mutation rate under
which almost surely there are exceptional times at which is a
finite sum of weighted Dirac masses. In the present work we discuss the
existence of such exceptional times for the generalized Fleming-Viot processes.
In the case of Beta-Fleming-Viot processes with index we
show that - irrespectively of the mutation rate and - the number of
atoms is almost surely always infinite. The proof combines a Pitman-Yor type
representation with a disintegration formula, Lamperti's transformation for
self-similar processes and covering results for Poisson point processes
Collision local time of transient random walks and intermediate phases in interacting stochastic systems
In a companion paper (M. Birkner, A. Greven, F. den Hollander, Quenched LDP for words in a letter sequence, Probab. Theory Relat. Fields 148, no. 3/4 (2010), 403-456), a quenched large deviation principle (LDP) has been established for the empirical process of words obtained by cutting an i.i.d. sequence of letters into words according to a renewal process. We apply this LDP to prove that the radius of convergence of the generating function of the collision local time of two independent copies of a symmetric and strongly transient random walk on Zd, d = 1, both starting from the origin, strictly increases when we condition on one of the random walks, both in discrete time and in continuous time. We conjecture that the same holds when the random walk is transient but not strongly transient. The presence of these gaps implies the existence of an intermediate phase for the long-time behaviour of a class of coupled branching processes, interacting diffusions, respectively, directed polymers in random environments
The effect of disorder on the free-energy for the Random Walk Pinning Model: smoothing of the phase transition and low temperature asymptotics
We consider the continuous time version of the Random Walk Pinning Model
(RWPM), studied in [5,6,7]. Given a fixed realization of a random walk Y$ on
Z^d with jump rate rho (that plays the role of the random medium), we modify
the law of a random walk X on Z^d with jump rate 1 by reweighting the paths,
giving an energy reward proportional to the intersection time L_t(X,Y)=\int_0^t
\ind_{X_s=Y_s}\dd s: the weight of the path under the new measure is exp(beta
L_t(X,Y)), beta in R. As beta increases, the system exhibits a
delocalization/localization transition: there is a critical value beta_c, such
that if beta>beta_c the two walks stick together for almost-all Y realizations.
A natural question is that of disorder relevance, that is whether the quenched
and annealed systems have the same behavior. In this paper we investigate how
the disorder modifies the shape of the free energy curve: (1) We prove that, in
dimension d larger or equal to three 3, the presence of disorder makes the
phase transition at least of second order. This, in dimension larger or equal
to 4, contrasts with the fact that the phase transition of the annealed system
is of first order. (2) In any dimension, we prove that disorder modifies the
low temperature asymptotic of the free energy.Comment: 18 page
Quenched large deviation principle for words in a letter sequence
When we cut an i.i.d. sequence of letters into words according to an
independent renewal process, we obtain an i.i.d. sequence of words. In the
\emph{annealed} large deviation principle (LDP) for the empirical process of
words, the rate function is the specific relative entropy of the observed law
of words w.r.t. the reference law of words. In the present paper we consider
the \emph{quenched} LDP, i.e., we condition on a typical letter sequence. We
focus on the case where the renewal process has an \emph{algebraic} tail. The
rate function turns out to be a sum of two terms, one being the annealed rate
function, the other being proportional to the specific relative entropy of the
observed law of letters w.r.t. the reference law of letters, with the former
being obtained by concatenating the words and randomising the location of the
origin. The proportionality constant equals the tail exponent of the renewal
process. Earlier work by Birkner considered the case where the renewal process
has an exponential tail, in which case the rate function turns out to be the
first term on the set where the second term vanishes and to be infinite
elsewhere. In a companion paper the annealed and the quenched LDP are applied
to the collision local time of transient random walks, and the existence of an
intermediate phase for a class of interacting stochastic systems is
established.Comment: 41 pages, 2 figures. Acronym LDP spelled out in title, main result
strengthened to cover more general "letter" spaces, application to collision
local times removed (this part will become a separate manuscript
Copolymer with pinning: variational characterization of the phase diagram
This paper studies a polymer chain in the vicinity of a linear interface
separating two immiscible solvents. The polymer consists of random monomer
types, while the interface carries random charges. Both the monomer types and
the charges are given by i.i.d. sequences of random variables. The
configurations of the polymer are directed paths that can make i.i.d.
excursions of finite length above and below the interface. The Hamiltonian has
two parts: a monomer-solvent interaction ("copolymer") and a monomer-interface
interaction ("pinning"). The quenched and the annealed version of the model
each undergo a transition from a localized phase (where the polymer stays close
to the interface) to a delocalized phase (where the polymer wanders away from
the interface). We exploit the approach developed in [5] and [3] to derive
variational formulas for the quenched and the annealed free energy per monomer.
These variational formulas are analyzed to obtain detailed information on the
critical curves separating the two phases and on the typical behavior of the
polymer in each of the two phases. Our main results settle a number of open
questions.Comment: 46 pages, 9 figure
Terahertz radiation driven chiral edge currents in graphene
We observe photocurrents induced in single layer graphene samples by
illumination of the graphene edges with circularly polarized terahertz
radiation at normal incidence. The photocurrent flows along the sample edges
and forms a vortex. Its winding direction reverses by switching the light
helicity from left- to right-handed. We demonstrate that the photocurrent stems
from the sample edges, which reduce the spatial symmetry and result in an
asymmetric scattering of carriers driven by the radiation electric field. The
developed theory is in a good agreement with the experiment. We show that the
edge photocurrents can be applied for determination of the conductivity type
and the momentum scattering time of the charge carriers in the graphene edge
vicinity.Comment: 4 pages, 4 figure, additional Supplemental Material (3 pages, 1
figure
Issues of Processing and Multiple Testing of SELDI-TOF MS Proteomic Data
A new data filtering method for SELDI-TOF MS proteomic spectra data is described. We examined technical repeats (2 per subject) of intensity versus m/z (mass/charge) of bone marrow cell lysate for two groups of childhood leukemia patients: acute myeloid leukemia (AML) and acute lymphoblastic leukemia (ALL). As others have noted, the type of data processing as well as experimental variability can have a disproportionate impact on the list of interesting proteins (see Baggerly et al. (2004)). We propose a list of processing and multiple testing techniques to correct for 1) background drift; 2) filtering using smooth regression and cross-validated bandwidth selection; 3) peak finding; and 4) methods to correct for multiple testing (van der Laan et al. (2005)). The result is a list of proteins (indexed by m/z) where average expression is significantly different among disease (or treatment, etc.) groups. The procedures are intended to provide a sensible and statistically driven algorithm, which we argue provides a list of proteins that have a significant difference in expression. Given no sources of unmeasured bias (such as confounding of experimental conditions with disease status), proteins found to be statistically significant using this technique have a low probability of being false positives
Limit theorems for weakly subcritical branching processes in random environment
For a branching process in random environment it is assumed that the
offspring distribution of the individuals varies in a random fashion,
independently from one generation to the other. Interestingly there is the
possibility that the process may at the same time be subcritical and,
conditioned on nonextinction, 'supercritical'. This so-called weakly
subcritical case is considered in this paper. We study the asymptotic survival
probability and the size of the population conditioned on non-extinction. Also
a functional limit theorem is proven, which makes the conditional
supercriticality manifest. A main tool is a new type of functional limit
theorems for conditional random walks.Comment: 35 page
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