59,262 research outputs found
Space-time random walk loop measures
In this work, we investigate a novel setting of Markovian loop measures and
introduce a new class of loop measures called Bosonic loop measures. Namely, we
consider loop soups with varying intensity (chemical potential in
physics terms), and secondly, we study Markovian loop measures on graphs with
an additional "time" dimension leading to so-called space-time random walks and
their loop measures and Poisson point loop processes. Interesting phenomena
appear when the additional coordinate of the space-time process is on a
discrete torus with non-symmetric jump rates. The projection of these
space-time random walk loop measures onto the space dimensions is loop measures
on the spatial graph, and in the scaling limit of the discrete torus, these
loop measures converge to the so-called [Bosonic loop measures]. This provides
a natural probabilistic definition of [Bosonic loop measures]. These novel loop
measures have similarities with the standard Markovian loop measures only that
they give weights to loops of certain lengths, namely any length which is
multiple of a given length which serves as an additional
parameter. We complement our study with generalised versions of Dynkin's
isomorphism theorem (including a version for the whole complex field) as well
as Symanzik's moment formulae for complex Gaussian measures. Due to the lacking
symmetry of our space-time random walks, the distributions of the occupation
time fields are given in terms of complex Gaussian measures over complex-valued
random fields ([B92,BIS09]. Our space-time setting allows obtaining quantum
correlation functions as torus limits of space-time correlation functions.Comment: 3 figure
Monotonicity of the logarithmic energy for random matrices
It is well-known that the semi-circle law, which is the limiting distribution
in the Wigner theorem, is the minimizer of the logarithmic energy penalized by
the second moment. A very similar fact holds for the Girko and
Marchenko--Pastur theorems. In this work, we shed the light on an intriguing
phenomenon suggesting that this functional is monotonic along the mean
empirical spectral distribution in terms of the matrix dimension. This is
reminiscent of the monotonicity of the Boltzmann entropy along the Boltzmann
equation, the monotonicity of the free energy along ergodic Markov processes,
and the Shannon monotonicity of entropy or free entropy along the classical or
free central limit theorem. While we only verify this monotonicity phenomenon
for the Gaussian unitary ensemble, the complex Ginibre ensemble, and the square
Laguerre unitary ensemble, numerical simulations suggest that it is actually
more universal. We obtain along the way explicit formulas of the logarithmic
energy of the mentioned models which can be of independent interest
Multivariate limit theorems in the context of long-range dependence
We study the limit law of a vector made up of normalized sums of functions of
long-range dependent stationary Gaussian series. Depending on the memory
parameter of the Gaussian series and on the Hermite ranks of the functions, the
resulting limit law may be (a) a multivariate Gaussian process involving
dependent Brownian motion marginals, or (b) a multivariate process involving
dependent Hermite processes as marginals, or (c) a combination. We treat cases
(a), (b) in general and case (c) when the Hermite components involve ranks 1
and 2. We include a conjecture about case (c) when the Hermite ranks are
arbitrary
Space-time random walk loop measures
In this work, we introduce and investigate two novel classes of loop measures, space–time Markovian loop measures and Bosonic loop measures, respectively. We consider loop soups with intensity (chemical potential in physics terms), and secondly, we study Markovian loop measures on graphs with an additional “time” dimension leading to so-called space–time random walks and their loop measures and Poisson point loop processes. Interesting phenomena appear when the additional coordinate of the space–time process is on a discrete torus with non-symmetric jump rates. The projection of these space–time random walk loop measures onto the space dimensions is loop measures on the spatial graph, and in the scaling limit of the discrete torus, these loop measures converge to the so-called Bosonic loop measures. This provides a natural probabilistic definition of Bosonic loop measures. These novel loop measures have similarities with the standard Markovian loop measures only that they give weights to loops of certain lengths, namely any length which is multiple of a given length which serves as an additional parameter. We complement our study with generalised versions of Dynkin’s isomorphism theorem (including a version for the whole complex field) as well as Symanzik’s moment formulae for complex Gaussian measures. Due to the lacking symmetry of our space–time random walks, the distributions of the occupation time fields are given in terms of complex Gaussian measures over complex-valued random fields [8], [10]. Our space–time setting allows obtaining quantum correlation functions as torus limits of space–time correlation functions
Limit theory for geometric statistics of point processes having fast decay of correlations
Let be a simple,stationary point process having fast decay of
correlations, i.e., its correlation functions factorize up to an additive error
decaying faster than any power of the separation distance. Let be its restriction to windows . We consider the statistic where denotes a score function
representing the interaction of with respect to . When depends
on local data in the sense that its radius of stabilization has an exponential
tail, we establish expectation asymptotics, variance asymptotics, and CLT for
and, more generally, for statistics of the re-scaled, possibly
signed, -weighted point measures , as . This gives the
limit theory for non-linear geometric statistics (such as clique counts,
intrinsic volumes of the Boolean model, and total edge length of the
-nearest neighbors graph) of -determinantal point processes having
fast decreasing kernels extending the CLTs of Soshnikov (2002) to non-linear
statistics. It also gives the limit theory for geometric U-statistics of
-permanental point processes and the zero set of Gaussian entire
functions, extending the CLTs of Nazarov and Sodin (2012) and Shirai and
Takahashi (2003), which are also confined to linear statistics. The proof of
the central limit theorem relies on a factorial moment expansion originating in
Blaszczyszyn (1995), Blaszczyszyn, Merzbach, Schmidt (1997) to show the fast
decay of the correlations of -weighted point measures. The latter property
is shown to imply a condition equivalent to Brillinger mixing and consequently
yields the CLT for via an extension of the cumulant method.Comment: 62 pages. Fundamental changes to the terminology including the title.
The earlier 'clustering' condition is now introduced as a notion of mixing
and its connection to Brillinger mixing is remarked. Newer results for
superposition of independent point processes have been adde
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