Oberseminar Stochastik

Mittwoch, 03. September 2014

14:15  14:55 Uhr: Kathrin Skubch (Universität Frankfurt)
Pfadlängen in scale free trees
Abstract:
Masterarbeit
15:00  15:40 Uhr: Andrea Kuntschik (Universität Frankfurt)
Konvergenzraten für PolyaUrnen
Abstract:
Masterarbeit
Raum 711 (klein), RobertMayerStr. 10



Oberseminar Stochastik

Freitag, 25. Juli 2014

10:15 Uhr: Stefan Albert (Universität Frankfurt)
Ein MultipleFilterTest zur Detektion von Varianzänderungen in Erneuerungsprozessen
Abstract:
Masterarbeit
11:15 Uhr: Benjamin Straub (Universität Frankfurt)
Optimierung von Phasen und Ratenparametern in einem stochastischen Modell neuronaler Feueraktivität
Abstract:
Masterarbeit
Raum 110, RobertMayerStr. 10

Oberseminar Stochastik

Montag, 21. Juli 2014

14:15 Uhr: Carolin Breitenbach (Universität Frankfurt)
Asymptotik der totalen Astlänge im XiKoaleszenten mit Staub
Abstract:
Masterarbeit
Raum 711 (klein), RobertMayerStr. 10

Oberseminar Stochastik

Dienstag, 15. Juli 2014

10:15 Uhr  14 Uhr: Gaby Schneider & Team (Universität Frankfurt)
Abschlusspräsentation des Statistischen Praktikums
Programm mit den Abstracts:
http://www.math.unifrankfurt.de/~ismi/schneider/StatPrakt14.html
Raum 711 (klein), RobertMayerStr. 10

Stochastisches Kolloquium

Mittwoch, 09. Juli 2014

14:15 Uhr: Matthias Meiners (TU Darmstadt)
Solutions to multivariate smoothing equations
Abstract:
In several models of applied probability such as cyclic Pólya urns, mary
search trees, and fragmentation processes, limiting distributions of
quantities of interest are solutions to smoothing equations in the complex
plane. Further, the stationary solutions of certain 3dimensional
kinetictype evolution equations satisfy smoothing equations with random
similarities as coefficients.
In my talk, I will consider smoothing equations in dimension d with
random similarities as coefficients. This is a unified framework which
contains all examples listed above. The main focus of the talk is on
the problem of determining all solutions to these equations and to
compare the set of solutions with the corresponding ones in one
dimension.
The talk is based on ongoing joint research with Sebastian Mentemeier
(Wroclaw).
Raum 711 (klein), RobertMayerStr. 10

RheinMain Kolloquium Stochastik

Freitag, 27. Juni 2014

15:00 Uhr: Markus Heydenreich (Leiden)
Spontaneous breaking of rotational symmetry in the presence of defects
Abstract:
The formation of crystals, and in particular melting and freezing transitions, are not yet mathematically
understood. It is expected that crystallization phenomena are intricately connected with the breaking of
certain symmetries. In this talk, we consider a simple twodimensional model of crystallization with random
defects in thermal equilibrium. We prove a strong form of spontaneous breaking of rotational symmetry at low
temperatures.
(Based on joint work with Franz Merkl and Silke Rolles).
16:30 Uhr: Erwin Bolthausen (Zürich)
Exit distributions for random walks in anisotropic
random environments (joint work with Erich Baur and Ofer Zeitouni)
Abstract:
Most results on nonballistic (and nonreversible) random walks in random environments
use an isotropy condition for the random environment. We present a recent joint result with
Erich Baur on an anisotropic case in dimension 3 and above. We also discuss briefly the difficult
twodimensional case which is work in progress (with Erich Baur and Ofer Zeitouni).
TU Darmstadt, Fachbereich Mathematik, Schloßgartenstr. 7 (Gebäude S215), Raum: tba

Stochastisches Kolloquium

Donnerstag, 12. Juni 2014

16:15 Uhr: Stas Volkov (Universität Lund, Schweden)
5x+1: how many go down?
Abstract:
I will talk about how probabilistic methods of analyzing randomlylabeled trees can
provide an important insight on the 5x+1 version of the famous Collatz problem (3x+1).
Though no rigorous results about number theory will be proved in my talk, a number of
properties of the trees with random labels will be rigorously established. The work is
inspired by Yakov Sinai's earlier work on the statistic of 3x+1 problem.
Raum 308, RobertMayerStr. 68

RheinMain Kolloquium Stochastik

Freitag, 6. Juni 2014

15:15 Uhr: Volker Betz (TU Darmstadt)
Long cycles of spatial random permutations
Abstract:
Consider a locally finite set X of points in R^d, and the set S of permutations mapping X to X.
Roughly speaking, a model of spatial random permutations is a probability measure on S which
suppresses permutations where typical distances between points and their images are large.
Mathematically, such a measure has to be constructed via finite volume approximations, and
then the existence of an infinite volume limit has to be proved. In the first part of my talk
I will show how to do this when X is a regular lattice.
The most interesting feature of spatial random permutations is the presence of a phase transition
in dimension 3 and higher:
when the penalization of long distances between points and their images is strong enough, the
cycle C_x starting from a given point x is known to be finite with probability one. When the penalization
strength is below a critical value, it is believed (and seen numerically) that C_x is infinite with positive
probability. This phenomenon is of physical relevance since it is closely connected with condensation in the
interacting Bose gas. Mathematically, it is so far only understood in the annealed variant of the model.
In the second part of my talk, I will give an overview on what is known about the existence of infinite cycles.
16:45 Uhr: Steffen Dereich (Universität Münster)
Condensation in preferential attachment models with fitness
Abstract:
A popular model for complex networks is the preferential attachment model which gained popularity
in the end of the 90's since it gives a simple explanation for the appearance of power laws
in real world networks. Mathematically, one considers a sequence of random graphs that is
built dynamically according to a simple rule. In each step a new vertex is added and linked
randomly by a random or deterministic number of edges to the vertices already present in the system.
In this process, links to vertices with high degree are preferred. A variant of the model, additionally,
assigns each vertex a random positive fitness (say a $\mu$distributed value) which has a linear impact
on its attractivity in the network formation.
Such network models show a phase transition for compactly supported $\mu$.
In the condensation phase, in the limit, there is a comparably small set of vertices
(the condensate) that attracts a constant fraction of new links established by new vertices.
This condensation effect was observed for the first time by Bianconi and Bara\'asi in 2001, where
it was coined BoseEinstein phase due to similarities to BoseEinstein condensation. The fitness of the
vertices in the condensate gradually converges to the essential supremum of $\mu$ and in the talk we discuss
the dynamics of this process.
Universität Frankfurt, Institut für Mathematik, RobertMayerStr. 10, Raum 711 (groß), 7. Stock

RheinMain Kolloquium Stochastik

Freitag, 23. Mai 2014

15:15 Uhr: Jiri Cerny (Wien)
Vacant set of random walk on discrete torus
Abstract:
We prove a phase transition in the behaviour of certain macroscopic observable on the vacant set of random
walk on the ddimensional discrete torus, d>2. To this end we discuss a technique of coupling of Markov
chains so that their ranges almost coincide all the time.
This is a joint work with A. Teixeira (IMPA)
16:45 Uhr: Noam Berger (TU München)
Local CLT for ballistic random walk in random environment
Abstract:
We prove a local version of the quenched CLT for strongly ballistic random walk in random environment
in dimension 4 and higher.
Joint work with Ron Rosenthal and Moran Cohen.
Universität Mainz, Institut für Mathematik, Staudingerweg 9, Raum 05432 (Hilbertraum)

Stochastisches Kolloquium

Mittwoch, 23. März 2014

14:00 Uhr: Peter Ney (University of Wisconsin, Madison)
Large deviations of Markov chains
15:00 Uhr: Vladimir Vatutin (Steklov Institut, Moskau)
Macroscopic and microscopic structure of the family tree of decomposable branching process
Raum 711 (klein), RobertMayerStr. 10

Oberseminar Stochastik

Mittwoch, 12. März 2014

14:00 Uhr: Christina Diehl (Universität Frankfurt)
Stabile Verteilungen von Längen in BetaKoaleszenten
15:00 Uhr: Christopher P. Imanto (Universität Frankfurt)
Die Fixierungswahrscheinlichkeit eines LambdaWrightFisherProzesses mit Selektion
Raum 711 (groß), RobertMayerStr. 10

Stochastisches Kolloquium

Montag, 24. Februar 2014

12:45 Uhr: Ted Cox (Syracuse University)
Convergence of finite voter model densities
Raum 711 (groß), RobertMayerStr. 10

RheinMain Kolloquium Stochastik

Freitag, 31. Januar 2014

15:30 Uhr: Zhan Shi (Paris)
Biased random walks on trees
Abstract:
I am going to make some elementary discussions on asymptotic properties of randomlybiased random walks on
supercritical GaltonWatson trees, and present a couple of unanswered questions. Joint work with Yueyun Hu.
17:00 Uhr: Anton Bovier (Bonn)
Extremal Processes in Branching Brownian Motions
Abstract:
I review the constuction of the process of extremes of branching Brownian motion.
Using a time change between the branching mechanism and the Brownian motion, one can construct
a larger class of ``variable speed'' branching Brownian motions, introduced by Derrida and Spohn.
I present some recent results (obtained with Lisa Hartung) on the extremal processes for some cases.
TU Darmstadt, Fachbereich Mathematik, Schloßgartenstr. 7 (Gebäude S215), Raum 51

Stochastisches Kolloquium Extratermin

Donnerstag, 23. Januar 2014

14:30 Uhr: Dr. Tatjana Tchumatchenko und Sabrina Münzberg (Frankfurt)
Informationstheoretische Analyse von neuronalen Netzwerken
Abstract:
Unser Gehirn besteht aus einem Netzwerk von Neuronen, die untereinander über digitale Strompulse,
sogenannte Spikes, kommunizieren. Der Austausch von Spikes ermöglicht einen Austausch von Information.
Wir sind daran interessiert festzustellen, wie viel Information im Netzwerk kodiert ist.
Mit Hilfe informationstheoretischer Ansätze werden wir zunächst die Menge der Information analytisch bestimmen,
die eine SpikeAntwort über den eingehenden Stimulus eines Neurons enthält. Diese analytische Quantifizierung
basiert auf der Mathematik von LevelCrossings stochastischer Prozesse.
Der Vortrag wird sich auf den Fall des einzelnen Neurons konzentrieren und einen Ausblick auf die
Analyse des Netzwerkes geben.
Raum 109d, RobertMayerStr. 68

RheinMain Kolloquium Stochastik

Freitag, 13. Dezember 2013

13:30  14:30 Uhr: JeanFrançois Marckert (Bordeaux)
Compact convexes of the plane and probability theory
Abstract:
This is a common work with David Renault (LaBRI).
We revisit the connections between compact convexes
of the plane and probability measures.
First, we will discuss the existing models of random
compact convexes of the plane, and the lack of models
of smooth random convexes.
But, the main point we want to discuss is a deep
bijection attributed to GaussMinkowski (!), between the set of
probability measures $\mu$ on $[0,2\pi]$ such that $\int_0^{2\pi}
e^{ix}d\mu(x)=0$ and compact convexes of the plane with
length 1. This deep connection was not stated in terms of
probability theory,and was seen only from a geometrical
perspective.
Introducing probabilistic tools in this domain appeared to be really
powerful.
We show that some natural operations on convexes  for
example, the Minkowski sum  have natural translations in terms of
operations on probability measures. Further applications are
provided, as a new notion of convolution of convexes, and the proof
that a polygonal curve associated with a sample of $n$ random
variables (satisfying $\int_0^{2\pi} e^{ix}d\mu(x)=0$) converges to
a convex associated with $\mu$ at speed $\sqrt{n}$, result much
similar to the convergence of empirical process in statistics.
Finally, we present some models of smooth random convexes and
simulations.
14:30  15:00 Uhr: Kaffee
15:00  16:00 Uhr: Rudolf Grübel (Hannover)
From trees to functions to ultrametric spaces, and back
Abstract:
The famous Harris correspondence provides a very useful
link between simply generated random trees and random
functions on the unit interval. I will
 describe two attempts (2009, 2014) to obtain an
analogue for search trees,
 discuss some current work, some of it joint with
Steve Evans and Anton Wakolbinger, on the relation
to ordered ultrametric spaces and IDLA models.
Universität Frankfurt, Institut für Mathematik, RobertMayerStr. 10, Raum 110, 1. Stock

Stochastisches Kolloquium Extratermin

Dienstag, 10. Dezember 2013

14:15 Uhr: Peter Kratz (Marseille)
Anwendungen großer Abweichungen in der Epidemiologie
Abstract:
Wir betrachten Kompartimentmodelle zur Beschreibung der Ausbreitung von Krankheiten.
Es ist bekannt, dass deterministische und entsprechende stochastische Modelle dieser Art
über ein Gesetz der großen Zahlen miteinander verbunden sind. Wir betrachten große Abweichungen
des stochastischen Modells vom deterministichen Modell und erhalten ein Resultat über den
Austrittszeitpunkt des stochastischen Prozesses aus dem Anziehungsgebiet eines stabilen Gleichgewichts
des deterministischen Modells.
Angewandt auf konkrete epidemiologische Modelle erhalten wir dadurch zum Beispiel den Zeitpunkt an dem
eine Krankheit ausstirbt.
Raum 711 (klein), RobertMayerStr. 10

Stochastisches Kolloquium

Mittwoch, 27. November 2013

14:15 Uhr: Nicolas Broutin (INRIA Rocquencourt)
The scaling limit of a minimum spanning tree of a complete graph
Abstract:
Consider the minimum spanning tree of the complete graph with $n$ vertices, when edges are assigned
independent random weights. We show that, when this tree is endowed with the graph distance renormalized
by $n^{1/3}$ and with the uniform measure on its vertices, the resulting space converges in distribution
as $n\to\infty$ to a random limiting metric space in the GromovHausdorffProkhorov topology. We show
that this space is a random binary real tree and has Minkowski dimension three almost surely. In particular,
its law is mutually singular with that of the Brownian continuum random tree or any rescaled version thereof.
Our approach relies on a coupling between the minimum spanning tree problem and the ErdosRenyi random graph,
and the explicit description of its scaling limit in the socalled critical window.
Raum 711 (klein), RobertMayerStr. 10

RheinMain Kolloquium Stochastik

Freitag, 22. November 2013

15:15 Uhr: Martin Zerner (Tübingen)
On the forward branching process for onedimensional excited random walks
Abstract:
We consider onedimensional excited random walks in random environments and explain how their relation to certain
branching processes with migration can be used to obtain various recurrence/transience properties of these walks.
This talk is partially based on joint work with Elena Kosygina.
16:1516:45 Uhr: Kaffee und Tee
16:45 Uhr: Artem Sapozhnikov (Leipzig)
Quenched invariance principle for simple random walk in correlated percolation models
Abstract:
Let S be a random subgraph of Z^d. I will discuss a set of conditions on the distribution of S under
which the quenched invariance principle holds for the simple random walk on S. Examples of models satisfying
the conditions include Bernoulli percolation, random interlacements and its vacant set, level sets of the
Gaussian free field.
This is a joint work with E. Procaccia and R. Rosenthal, based on an earlier joint work with A. Drewitz and B. Ráth.
Im Anschluss gemeinsame Nachsitzung
Universität Mainz, Institut für Mathematik, Staudingerweg 9, Raum 05432 (Hilbertraum)

Stochastisches Kolloquium

Mittwoch, 21. August 2013

14:15 Uhr: Markus Kuba (Wien)
Urnenmodelle mit Mehrfachziehungen
Abstract:
tba.
Raum 711 (klein), RobertMayerStr. 10

Oberseminar Stochastik

Dienstag, 13. August 2013

14:15 Uhr: Benjamin Dadoun (ENS Cachan)
A statistical view on exchanges in Quickselect
Abstract:
In this talkt we study the number of key exchanges required by Hoare's FIND algorithm (also called Quickselect)
when operating on a uniformly distributed random permutation and selecting an independent uniformly distributed
rank. After normalization we give a limit theorem where the limit law is a perpetuity characterized by a recursive
distributional equation. To make the limit theorem useable for statistical methods and statistical experiments we
provide an explicit rate of convergence in the KolmogorovSmirnov metric, a numerical table of the limit law's
distribution function and an algorithm for exact simulation from the limit distribution. We also investigate the
limit law's density. This case study provides a program applicable to other cost measures, alternative models for
the rank and more balanced choices of the pivot element such as medianof$2t+1$ versions of Quickselect.
Raum 711 (groß), RobertMayerStr. 10

Stochastisches Kolloquium

Montag, 15. Juli 2013

14:00 Uhr: Vi LE (Universität Marseille)
Height and total mass of a forest of genealogical trees with competition
Abstract:
We consider a discrete model of population with interaction where the birth and death rates are non linear functions
of the population size. After proceeding to renormalization of the model parameters, we obtain in the limit of large
population that the population size evolves as a diffusion solution of the SDE
${Z}_{\mathrm{t}}^{\mathrm{x}}=\; x\; +{\int}_{0}^{\mathrm{t}}f({Z}_{\mathrm{s}}^{\mathrm{x}})\; ds\; +\; 2{\int}_{0}^{\mathrm{t}}\sqrt{{Z}_{\mathrm{s}}^{\mathrm{x}}}{dW}_{\mathrm{s}}^{\mathrm{x}}$.
There is a natural way of describing the genealogical tree of the discrete
population. The genealogy in the continuous limit is described by a real tree
(in the sense of Aldous). In both the discrete and the continuous case, we
study the height and the total mass of the genealogical tree, as an (increasing)
function of the initial population. We show that the expectation of the height
of the tree remains bounded as the size of the initial population tends to
infinity iff
${\int}_{1}^{\mathrm{\infty}}1/\; f(x)\; dx\; <\; \infty $,
while the expectation of the total mass of the tree remains
bounded as the size of the initial population tends to infinity
iff
${\int}_{1}^{\mathrm{\infty}}x\; /\; f(x)\; dx\; <\; \infty $.
Raum 711 (klein), RobertMayerStr. 10

RheinMain Kolloquium Stochastik

Freitag, 12. Juli 2013

15:15 Uhr: Francesco Caravenna (Mailand)
Scaling limits and universality for random pinning models
Abstract:
We consider the socalled random pinning model, which may be described as a Markov chain that receives a random
reward/penalty each time it visits a given site. When the return time distribution of the Markov chain has a polynomial
tail, with exponent larger than 1/2, the model is said to be disorderrelevant, since an arbitrarily small amount of
external randomness (quenched disorder) changes radically the critical properties of the model. In this regime, we show
that the partition function of the model, under an appropriate weak coupling scaling limit, converges to a universal
quantity, given by an explicit Wiener chaos expansion. This quantity can be viewed as the partition function of a
universal "continuum random pinning model", whose construction is part of our approach.
(Joint work with Nikos Zygouras and Rongfeng Sun)
16:45 Uhr: Dima Ioffe (Technion Haifa + Universität Bonn)
An invariance principle for random walks with prewetting
Abstract:
I shall discuss an invariance principle for a class of 1+1 SOS interfaces (walks) in potential fields.
The interfaces are constrained to stay above the wall. The potential field is \lambda X^a with a\geq 1. The case a=1
corresponds to tilting areas below interfaces. Limiting objects, as tends to zero, are reversible diffusions with
drifts given by logderivatives of ground states for the associated singular SturmLiouville operators. For a=1 such ground state
is a rescaled Airy function, and the corresponding diffusion was derived by Ferrari and Spohn in the context of scaling
limits for Brownian motions conditioned to stay above circular barriers.
Based on a work in progress with Senya Shlosman and Yvan Velenik.
Universität Mainz, Institut für Mathematik, Staudingerweg 9, Raum 05432 (Hilbertraum)

Oberseminar Stochastik

Mittwoch, 03. Juli 2013

14:15 Uhr: Franziska Wandtner (Universität Frankfurt)
Simulationen und ihre Rolle für das Verständnis von Stochastik
Abstract:
tba.
Raum 308, RobertMayerStr. 68

Stochastisches Kolloquium

Mittwoch, 26. Juni 2013

14:15 Uhr: Henning Sulzbach (Universität Frankfurt)
Gaussians limits for improved Quickselect variants
Abstract:
Despite of its weak worst case behaviour, Quickselect (or FIND) has become one of the most popular selection algorithms.
We discuss and analyse (in a certain sense) optimal Medianofk Quickselect routines with increasing subsample size k introduced
by Martinez and Roura (2001/02). Here, the complexity of the algorithm measured in terms of the number of key comparisons converges
in the Skorokhod space to a Gaussian process with interesting covariance function. We discuss several properties of the limit process.
Our results are based on a new approach relying on the contraction method towards functional limit theorems for processes satisfying
distributional recurrences where uniform convergence can not be expected and jumps persist in the limit.
If time permits, we will also consider (in a different sense) optimal adapted Quickselect variants where no results beyond asymptotic
expansions of first order are known.
Raum 308, RobertMayerStr. 68

Mathematisches Kolloquium

Donnerstag, 20. Juni 2013

ab 16:00 Uhr: Tee
16:15 Uhr: Steven N. Evans (Departments of Statistics and Mathematics, University of California at Berkeley. U.S.A.)
Optimal transport of measures and metagenomics
Abstract:
Metagenomics attempts to sample and study all the genetic
material present in a community of microorganisms in environments
that range from the human gut to the open ocean. This enterprise is
made possible by highthroughput pyrosequencing technologies that
produce a ``soup'' of DNA fragments which are not a priori associated
with particular organisms or with particular locations on the genome.
Statistical methods can be used to assign these fragments to locations
on a reference phylogenetic tree using preexisting information about
the genomes of previously identified species. Each metagenomic sample
thus results in a cloud of points on the reference tree. In seeking to
answer questions such as what distinguishes the vaginal microbiomes
of women with bacterial vaginosis from those of woment who don't,
one is led to consider statistical methods
for distinguishing between such clouds. I will discuss joint work
with Erick Matsen from the Fred Hutchinson Cancer Research Center
in which we use ideas based on distances between probability measures
that go back to Gaspard Monge's 1781 ``M\'emoir sur la th\'eorie
des d\'eblais et des remblais'' as well as some familiar objects
(e.g. reproducing kernel Hilbert spaces) from the world of
Gaussian processes.
Raum 302 (HilbertRaum), RobertMayerStr. 10

Stochastisches Kolloquium

Mittwoch, 19. Juni 2013

14:15 Uhr: Christian Böinghoff (Universität Frankfurt)
Conditional Limit Theorems for Intermediately Subcritical Branching Processes in Random Environment
Abstract:
Branching processes in random environment (BPRE) are a model for the development of a population of individuals, which are exposed to
a random environment. More precisely, it is assumed that the offspring distribution of the individuals varies in a random fashion,
independently from one generation to the other. Given the environment, all individuals reproduce independently according to
the same mechanism.
As it turns out, there is a phase transition within the subcritical regime, i.e. the asymptotics of the survival probability and the
behavior of the branching process, conditioned on survival, changes fundamentally. In the talk, we will describe the behavior of
the BPRE in the intermediately subcritical case, which is at the borderline of the phase transition. In this case, the BPRE conditioned
on survival consists of periods with supercritical growth, alternating with subcritical periods of small population sizes.
This kind of 'bottleneck' behavior appears under the annealed approach only in the intermediately subcritical case. Our results
include the asymptotics of the survival probability and conditional limit theorems for the (properly scaled) random environment and
the BPRE, conditioned on nonextinction.
The proofs rely on a construction of the conditioned branching tree going back to Geiger (1999) and Lyons, Pemantle and Peres (1995). Using this
construction and a conditional limit theorem for the (properly scaled) random environment, simulations are feasible in the case
of geometric offspring distributions. These are used to illustrate our results.
15:15 Uhr: Vincent Bansaye (Ecole Polytechnique, Palaiseau)
On the extinction of continuous state branching processes with
catastrophes
Abstract:
We consider continuous state branching processes (CSBP's for short) with additional multiplicative jumps, which we call
catastrophes. Informally speaking, the dynamics of the CSBP is perturbed by
independent random catastrophes which cause negative (or positive) jumps to the original process. These jumps
are described by a Lévy process with paths of bounded variation.
We construct this class of processes CSBP in random environment as the unique solution of a SDE and
characterize their Laplace exponent as the solution of a backward ODE. We can then study their asymptotic behavior
and establish whether the process becomes extinct. For a class of processes for which extinction and absorption coincide,
we determine the speed of extinction of the process.
Finally, we apply these results to a cell infection model, which was a motivation for considering such
CSBP's with catastrophes. If some time remains, we will speak a little about a more general class of CSBP in random
environment.
Raum 308, RobertMayerStr. 68

Oberseminar Stochastik

Mittwoch, 12. Juni 2013

14:15 Uhr: Götz Kersting (Universität Frankfurt)
Das Ranking von Webseiten bei Google (Probevortrag)
Probevortrag für die Night of Science.
Raum 308, RobertMayerStr. 68

Stochastisches Kolloquium Extratermin

Mittwoch, 12. Juni 2013

10:15 Uhr: Steven N. Evans (Departments of Statistics and Mathematics, University of California at Berkeley, U.S.A.)
Unseparated pairs and fixed points in random permutations
Abstract:
``Smoosh shuffling'' is a naive physical mechanism for randomizing a deck of cards. In order to measure the effectiveness of smoosh shuffling,
Persi Diaconis and his collaborators counted the number of cards in the shuffled deck that are still immediately followed by their successor in
the original deck and compared the observed distribution of this statistic with the corresponding distribution derived from computer simulations
of perfect shuffles. It turns out that the distribution of this statistic was already considered by William Allen Whitworth in his 1867 book
``Choice and Chance'', where he computed the probability that there are no such cards. I will show how it is possible to derive the explicit
distribution of the statistic using a variety of methods ranging from elementary enumeration, through coupled constructions of random permutations
using the ``Chinese restaurant process'', to combinatorial bijections using the {\em transformation fondamentale} of Foata and Sch{\"u}tzenberger.
Some natural extensions of this problem lead to questions about the asymptotics of the number of fixed points of the commutator of a given permutation
with a uniformly distributed random one. I will show how such problems can be solved using a relatively simple instance of the exchangeable pairs
version of Stein's method. This is joint work with Persi Diaconis (Stanford University) and Ron Graham (University of California at San Diego).
Raum 302 (HilbertRaum), RobertMayerStr. 10

Stochastisches Kolloquium

Mittwoch, 29. Mai 2013

14:15 Uhr: Linglong Yuan (Université Paris 13, Villetaneuse)
Smalltime behavior of Beta ncoalescent
Abstract:
Beta$(2\alpha, \alpha)$ ncoalescent with $1\le\alpha\le2$ serves as an important tool to model the genealogy
of a sample of $n$ individuals from a large population which allows one individual to have many children in the
next generation with a large probability. A good understanding of the "shape" of this process helps the biologists
to well describe the genealogical trees. This talk will give some results to know the behavior of the process at
"small times". The "small time" here is of the order of one uniformly chosen external branch length, which is
therefore the first object to study. More generally, we will pick $k (k\geq1)$ individuals randomly and see their
asymptotic external branch lengths. At the moment of the coalescence of individual 1, the quantity so called "minimal
clade size" in biology which is the total number of offspring at time $0$ of those individuals coalesced has been proved
to have a limit theorem. Moreover, the largest total number of offspring at time $0$ of one individual at that moment
enjoys also a nice limit behavior. Another interesting quantity is the probability for individual $1$ to be coalesced
with $k$ individuals for any $k\geq 1$.
The proofs of these results rely heavily on the connections between Beta$(2\alpha, \alpha)$ ncoalescent and
Beta$(2\alpha, \alpha)$coalescent. The latter can be considered as the limit of the former when $n$ tends to $+\infty$.
Beta$(2\alpha, \alpha)$coalescent has been well studied thanks to its rich relations with other processes, such as the
Beta flemingviot process, the ranked coalescent process, CSBP, etc. The usage of the connections to get properties of
Beta$(2\alpha, \alpha)$ ncoalescent from Beta$(2\alpha, \alpha)$coalescent seems new and this point of view can be
applied to other coalescent processes.
Raum 308, RobertMayerStr. 68

RheinMain Kolloquium Stochastik

Freitag, 24. Mai 2013

15:15 Uhr: JeanFrançois Le Gall (Université ParisSud)
The Brownian map: A universal model of random geometry
Abstract:
Consider a triangulation of the sphere chosen uniformly at random among all triangulations with a fixed
number of faces (two triangulations are identified if they correspond via an orientationpreserving homeomorphism of the sphere).
We equip the vertex set of this triangulation with the usual graph distance. When the number of faces tends to infinity, the
(suitably rescaled) resulting metric space converges in distribution, in the sense of the GromovHausdorff distance, towards a
random compact metric space called the Brownian map. This result, which confirms a conjecture of Schramm in 2006, holds with the
same limit for much more general random graphs drawn on the sphere, including the socalled quadrangulations of the sphere. The
Brownian map thus appears as a universal model of a random surface, which is homeomorphic to the sphere but has Hausdorff dimension
4. We will discuss the construction of the Brownian map, and present some ingredients of the proof of our main convergence result.
The behavior of GaltonWatson trees whose offspring distribution is critical and has finite variance, conditioned on having a fixed
large size, has drawn a lot of attention. We will be interested in what happens outside of this typical framework. More precisely,
what can be said if the conditioning on having a fixed size is replaced by the conditioning on having a fixed number of leaves? What
happens if the offspring distribution is not critical?
16:45 Uhr: Alison Etheridge (University of Oxford)
Modelling natural selection
Abstract:
The basic challenge of mathematical population genetics is to understand the relative importance of the different
forces of evolution in shaping the genetic diversity that we see in the world around us. This is a problem that has
been around for a century, and a great deal is known. However, a proper understanding of the role of a population's
spatial structure is missing. Recently we introduced a new framework for modelling populations that evolve in a spatial
continuum. In this talk we briefly describe this framework before outlining some preliminary results on the importance
of spatial structure for natural selection.
Universität Frankfurt, Institut für Mathematik, RobertMayerStr. 10, Raum 711 (groß), 7. Stock

Oberseminar Stochastik

Mittwoch, 22. Mai 2013

14:15 Uhr: Michael Messer (Universität Frankfurt)
A multiple filter test for change point detection in renewal processes with varying variance
Abstract:
Nonstationarity of the event rate is a persistent problem in modeling time series of events, such as neuronal spike trains.
Motivated by a variety of patterns in neurophysiological spike train recordings, we define a general class of renewal processes. This class
is used to test the null hypothesis of stationary rate versus a wide alternative of renewal processes with finitely many rate changes
(change points). Our test extends ideas from the filtered derivative approach by using multiple moving windows simultaneously.
To adjust the rejection threshold of the test we use a Gaussian process, which emerges as the limit of a cadlag filtered derivative process.
We analyze the benefits of our multiple filter test and study the increase in test power against a single window test. We also develop a multiple
filter algorithm, which can be used when the null hypothesis is rejected in order to estimate the number and location of change points. In a
sample data set of spike trains recorded from dopamine midbrain neurons in anesthetized mice in vivo, the detected change points agreed
closely with visual inspection, and in over 70% of all spike trains which were classified as rate nonstationary, different change
points were detected by different window sizes.
This work was supported by the LOEWESchwerpunkt `Neuronale Koordination Forschungsschwerpunkt Frankfurt'.
The manuscript is available at arXiv:1303.3594 [stat.AP].
Raum 308, RobertMayerStr. 68


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