10,105 research outputs found
On the oscillation rigidity of a Lipschitz function on a high-dimensional flat torus
Given an arbitrary -Lipschitz function on the torus ,
we find a -dimensional subtorus , parallel to the
axes, such that the restriction of to the subtorus is nearly a constant
function. The -dimensional subtorus is chosen randomly and uniformly. We
show that when , the
maximum and the minimum of on this random subtorus differ by at most
, with high probability.Comment: 8 page
Symmetric groups and checker triangulated surfaces
We consider triangulations of surfaces with edges painted three colors so
that edges of each triangle have different colors. Such structures arise as
Belyi data (or Grothendieck dessins d'enfant), on the other hand they enumerate
pairs of permutations determined up to a common conjugation. The topic of these
notes is links of such combinatorial structures with infinite symmetric groups
and their representations.Comment: 20p., 5 fi
Invariances in variance estimates
We provide variants and improvements of the Brascamp-Lieb variance inequality
which take into account the invariance properties of the underlying measure.
This is applied to spectral gap estimates for log-concave measures with many
symmetries and to non-interacting conservative spin systems
Modified log-Sobolev inequalities and two-level concentration
We consider a generic modified logarithmic Sobolev inequality (mLSI) of the
form for some difference operator , and show how it implies
two-level concentration inequalities akin to the Hanson--Wright or Bernstein
inequality. This can be applied to the continuous (e.\,g. the sphere or bounded
perturbations of product measures) as well as discrete setting (the symmetric
group, finite measures satisfying an approximate tensorization property,
\ldots).
Moreover, we use modified logarithmic Sobolev inequalities on the symmetric
group and for slices of the hypercube to prove Talagrand's convex
distance inequality, and provide concentration inequalities for locally
Lipschitz functions on . Some examples of known statistics are worked out,
for which we obtain the correct order of fluctuations, which is consistent with
central limit theorems
Invariant measures on multimode quantum Gaussian states
We derive the invariant measure on the manifold of multimode quantum Gaussian
states, induced by the Haar measure on the group of Gaussian unitary
transformations. To this end, by introducing a bipartition of the system in two
disjoint subsystems, we use a parameterization highlighting the role of
nonlocal degrees of freedom -- the symplectic eigenvalues -- which characterize
quantum entanglement across the given bipartition. A finite measure is then
obtained by imposing a physically motivated energy constraint. By averaging
over the local degrees of freedom we finally derive the invariant distribution
of the symplectic eigenvalues in some cases of particular interest for
applications in quantum optics and quantum information.Comment: 17 pages, comments are welcome. v2: presentation improved and typos
corrected. Close to the published versio
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