7,495 research outputs found
Quasi-stationary regime of a branching random walk in presence of an absorbing wall
A branching random walk in presence of an absorbing wall moving at a constant
velocity undergoes a phase transition as the velocity of the wall
varies. Below the critical velocity , the population has a non-zero
survival probability and when the population survives its size grows
exponentially. We investigate the histories of the population conditioned on
having a single survivor at some final time . We study the quasi-stationary
regime for when is large. To do so, one can construct a modified
stochastic process which is equivalent to the original process conditioned on
having a single survivor at final time . We then use this construction to
show that the properties of the quasi-stationary regime are universal when
. We also solve exactly a simple version of the problem, the
exponential model, for which the study of the quasi-stationary regime can be
reduced to the analysis of a single one-dimensional map.Comment: 2 figures, minor corrections, one reference adde
Duality in interacting particle systems and boson representation
In the context of Markov processes, we show a new scheme to derive dual
processes and a duality function based on a boson representation. This scheme
is applicable to a case in which a generator is expressed by boson creation and
annihilation operators. For some stochastic processes, duality relations have
been known, which connect continuous time Markov processes with discrete state
space and those with continuous state space. We clarify that using a generating
function approach and the Doi-Peliti method, a birth-death process (or discrete
random walk model) is naturally connected to a differential equation with
continuous variables, which would be interpreted as a dual Markov process. The
key point in the derivation is to use bosonic coherent states as a bra state,
instead of a conventional projection state. As examples, we apply the scheme to
a simple birth-coagulation process and a Brownian momentum process. The
generator of the Brownian momentum process is written by elements of the
SU(1,1) algebra, and using a boson realization of SU(1,1) we show that the same
scheme is available.Comment: 13 page
Discrete Feynman-Kac formulas for branching random walks
Branching random walks are key to the description of several physical and
biological systems, such as neutron multiplication, genetics and population
dynamics. For a broad class of such processes, in this Letter we derive the
discrete Feynman-Kac equations for the probability and the moments of the
number of visits of the walker to a given region in the phase space.
Feynman-Kac formulas for the residence times of Markovian processes are
recovered in the diffusion limit.Comment: 4 pages, 3 figure
Equivalence checking for weak bi-Kleene algebra
Pomset automata are an operational model of weak bi-Kleene algebra, which describes programs that can fork an execution into parallel threads, upon completion of which execution can join to resume as a single thread. We characterize a fragment of pomset automata that admits a decision procedure for language equivalence. Furthermore, we prove that this fragment corresponds precisely to series-rational expressions, i.e., rational expressions with an additional operator for bounded parallelism. As a consequence, we obtain a new proof that equivalence of series-rational expressions is decidable
Brzozowski goes concurrent - A kleene theorem for pomset languages
Concurrent Kleene Algebra (CKA) is a mathematical formalism to study programs that exhibit concurrent behaviour. As with previous extensions of Kleene Algebra, characterizing the free model is crucial in order to develop the foundations of the theory and potential applications. For CKA, this has been an open question for a few years and this paper makes an important step towards an answer. We present a new automaton model and a Kleene-like theorem that relates a relaxed version of CKA to series-parallel pomset languages, which are a natural candidate for the free model. There are two substantial differences with previous work: from expressions to automata, we use Brzozowski derivatives, which enable a direct construction of the automaton; from automata to expressions, we provide a syntactic characterization of the automata that denote valid CKA behaviours
Equivalence checking for weak bi-kleene algebra∗
Pomset automata are an operational model of weak bi-Kleene algebra, which describes programs that can fork an execution into parallel threads, upon completion of which execution can join to resume as a single thread. We characterize a fragment of pomset automata that admits a decision procedure for language equivalence. Furthermore, we prove that this fragment corresponds precisely to series-rational expressions, i.e., rational expressions with an additional operator for bounded parallelism. As a consequence, we obtain a new proof that equivalence of series-rational expressions is decidable
Anderson transition on the Cayley tree as a traveling wave critical point for various probability distributions
For Anderson localization on the Cayley tree, we study the statistics of
various observables as a function of the disorder strength and the number
of generations. We first consider the Landauer transmission . In the
localized phase, its logarithm follows the traveling wave form where (i) the disorder-averaged value moves linearly
and the localization length
diverges as with (ii) the
variable is a fixed random variable with a power-law tail for large with , so that all
integer moments of are governed by rare events. In the delocalized phase,
the transmission remains a finite random variable as , and
we measure near criticality the essential singularity with . We then consider the
statistical properties of normalized eigenstates, in particular the entropy and
the Inverse Participation Ratios (I.P.R.). In the localized phase, the typical
entropy diverges as with , whereas it grows
linearly in in the delocalized phase. Finally for the I.P.R., we explain
how closely related variables propagate as traveling waves in the delocalized
phase. In conclusion, both the localized phase and the delocalized phase are
characterized by the traveling wave propagation of some probability
distributions, and the Anderson localization/delocalization transition then
corresponds to a traveling/non-traveling critical point. Moreover, our results
point towards the existence of several exponents at criticality.Comment: 28 pages, 21 figures, comments welcom
Correlation Functions for an Elastic String in a Random Potential: Instanton Approach
We develop an instanton technique for calculations of correlation functions
characterizing statistical behavior of the elastic string in disordered media
and apply the proposed approach to correlations of string free energies
corresponding to different low-lying metastable positions. We find high-energy
tails of correlation functions for the case of long-range disorder (the
disorder correlation length well exceeds the characteristic distance between
the sequential string positions) and short-range disorder with the correlation
length much smaller then the characteristic string displacements. The former
case refers to energy distributions and correlations on the distances below the
Larkin correlation length, while the latter describes correlations on the large
spatial scales relevant for the creep dynamics.Comment: 5 pages; 1 .eps figure include
How Many Topics? Stability Analysis for Topic Models
Topic modeling refers to the task of discovering the underlying thematic
structure in a text corpus, where the output is commonly presented as a report
of the top terms appearing in each topic. Despite the diversity of topic
modeling algorithms that have been proposed, a common challenge in successfully
applying these techniques is the selection of an appropriate number of topics
for a given corpus. Choosing too few topics will produce results that are
overly broad, while choosing too many will result in the "over-clustering" of a
corpus into many small, highly-similar topics. In this paper, we propose a
term-centric stability analysis strategy to address this issue, the idea being
that a model with an appropriate number of topics will be more robust to
perturbations in the data. Using a topic modeling approach based on matrix
factorization, evaluations performed on a range of corpora show that this
strategy can successfully guide the model selection process.Comment: Improve readability of plots. Add minor clarification
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