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Random graphs with given degree distributions (Svante Janson, Uppsala University)
One important class of random graphs is given by taking a uniformly
random graph among all graphs with a given degree sequence.
An important example is a uniformly random r-regular graph with n
vertices for some given r and n. One standard method to study such random
graph uses the "configuration model" (Bollobas, 1980), which in general
generates a multigraph that may contain loops and multiple edges.
However, conditioning this random multigraph on being simple yields the
desired random graph.
I will talk about an alternative to this conditioning,
where instead we adjust the multigraph by local switchings to become simple.
This will not yield a uniform distribution, but almost.
This too is an old idea, but I will give new results, and some
applications to limit theorems.
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Testing independence of random elements with the distance
covariance (Thomas Mikosch, University of Copenhagen)
Distance covariance was introduced by Székely, Rizzo and Bakirov
(2007) as a measure of dependence between vectors of possibly distinct
dimensions. Since then it has attracted attention
in various fields of statistics and applied probability. The distance
covariance of two random vectors X and Y is a weighted L^2 distance
between the joint characteristic function of (X,Y) and the product
of the characteristic functions of X and Y. It has
the desirable property that it
is zero if and only if X and Y are independent. This is
in contrast to classical measures of dependence such as the correlation between
two random variables: zero correlation corresponds to the absence
of linear dependence but does not give any information about other kinds of
dependencies.
We consider the distance covariance for stochastic processes X and Y defined
on some interval and having square integrable paths,
including Lévy processes, fractional Brownian, diffusions, stable processes, and many more. Since distance covariance is defined for vectors we
consider discrete approximations to X and Y. We show that sample versions
of the discretized distance covariance converge to zero if and only if
X and Y are independent. The sample distance covariance is a degenerate
V-statistic and, therefore, has rate of convergence which is much faster than
the classical √ n-rates. This fact also shows nicely in
simulation studies for independent X and Y in contrast to dependent X and Y.
- Coulomb Gas Ensembles with Local Interactions
(Tatyana Turova, University of Lund)
We consider the system of particles on a finite interval with pair-wise
interaction and external force. We take into account interactions between nearest,
second nearest, and so on, up to K-th nearest neighbours, where K≥1 is arbitrary
but fixed.
This model was introduced by Malyshev (2015) to study the flow of charged particles
on a rigorous mathematical level. It is a simplified version of a 3-dimensional classical
Coulomb gas model. We study Gibbs distribution at finite positive temperature extending
results Malyshev and Zamyatin (2015) on the zero temperature case (ground states).
We derive in [3] the asymptotics for the mean and for the variances of the distances
between the neighbouring charges when K=1. It is proved that depending on the strength
of the external force there are several phase transitions in the local structure of the
configuration of the particles in the limit when the number of particles goes to infinity.
We identify 5 different phases for any positive temperature. The proof relies on a
conditional central limit theorem for non-identical independent random variables.
To study case K>1 we derive first the exponential decay of the correlations. This allows
to extend some results of [3] for a model with an arbitrary range of interactions.
[1] Malyshev, V. A.: Phase transitions in the one-dimensional Coulomb medium. Problems of Information Transmission, v. 51, no. 1, 31-36 (2015).
[2] Malyshev, V. A., and Zamyatin A. A.: One-Dimensional Coulomb Multiparticle Systems. Advances in Mathematical Physics, (2015).
[3] Turova, T. S. Phase transitions in the one-dimensional Coulomb gas ensembles. Ann. Appl. Probab. 28 (2018), no. 2, 1249-1291.
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Noise Stability, Noise Sensitivity and the argument Against Quantum Computers (Gil Kalai, Hebrew University of Jerusalem)
I will start with a gentle explanation of some basic notions about computation, and quantum computers.
Next I will present a theory of noise stability and noise sensitivity of various mathematical and physical systems and in
particular noisy intermediate scale quantum (NISQ) systems. Finally I present an argument for why quantum computers
are going to fail and what are the consequences.