@article{10405,
  abstract     = {We consider large non-Hermitian random matrices X with complex, independent, identically distributed centred entries and show that the linear statistics of their eigenvalues are asymptotically Gaussian for test functions having 2+ϵ derivatives. Previously this result was known only for a few special cases; either the test functions were required to be analytic [72], or the distribution of the matrix elements needed to be Gaussian [73], or at least match the Gaussian up to the first four moments [82, 56]. We find the exact dependence of the limiting variance on the fourth cumulant that was not known before. The proof relies on two novel ingredients: (i) a local law for a product of two resolvents of the Hermitisation of X with different spectral parameters and (ii) a coupling of several weakly dependent Dyson Brownian motions. These methods are also the key inputs for our analogous results on the linear eigenvalue statistics of real matrices X that are presented in the companion paper [32]. },
  author       = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J},
  issn         = {1097-0312},
  journal      = {Communications on Pure and Applied Mathematics},
  number       = {5},
  pages        = {946--1034},
  publisher    = {Wiley},
  title        = {{Central limit theorem for linear eigenvalue statistics of non-Hermitian random matrices}},
  doi          = {10.1002/cpa.22028},
  volume       = {76},
  year         = {2023},
}

@article{8500,
  abstract     = {The main model studied in this paper is a lattice of pendula with a nearest‐neighbor coupling. If the coupling is weak, then the system is near‐integrable and KAM tori fill most of the phase space. For all KAM trajectories the energy of each pendulum stays within a narrow band for all time. Still, we show that for an arbitrarily weak coupling of a certain localized type, the neighboring pendula can exchange energy. In fact, the energy can be transferred between the pendula in any prescribed way.},
  author       = {Kaloshin, Vadim and Levi, Mark and Saprykina, Maria},
  issn         = {0010-3640},
  journal      = {Communications on Pure and Applied Mathematics},
  keywords     = {Applied Mathematics, General Mathematics},
  number       = {5},
  pages        = {748--775},
  publisher    = {Wiley},
  title        = {{Arnol′d diffusion in a pendulum lattice}},
  doi          = {10.1002/cpa.21509},
  volume       = {67},
  year         = {2014},
}

@article{8517,
  abstract     = {We consider the evolution of a connected set on the plane carried by a space periodic incompressible stochastic flow. While for almost every realization of the stochastic flow at time t most of the particles are at a distance of order equation image away from the origin, there is a measure zero set of points that escape to infinity at the linear rate. We study the set of points visited by the original set by time t and show that such a set, when scaled down by the factor of t, has a limiting nonrandom shape.},
  author       = {Dolgopyat, Dmitry and Kaloshin, Vadim and Koralov, Leonid},
  issn         = {0010-3640},
  journal      = {Communications on Pure and Applied Mathematics},
  keywords     = {Applied Mathematics, General Mathematics},
  number       = {9},
  pages        = {1127--1158},
  publisher    = {Wiley},
  title        = {{A limit shape theorem for periodic stochastic dispersion}},
  doi          = {10.1002/cpa.20032},
  volume       = {57},
  year         = {2004},
}

@article{2731,
  abstract     = {We study the time evolution of a quantum particle in a Gaussian random environment. We show that in the weak coupling limit the Wigner distribution of the wave function converges to a solution of a linear Boltzmann equation globally in time. The Boltzmann collision kernel is given by the Born approximation of the quantum differential scattering cross section.},
  author       = {Erdös, László and Yau, Horng},
  issn         = {0010-3640},
  journal      = {Communications on Pure and Applied Mathematics},
  number       = {6},
  pages        = {667 -- 735},
  publisher    = {Wiley-Blackwell},
  title        = {{Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation}},
  doi          = {10.1002/(SICI)1097-0312(200006)53:6<667::AID-CPA1>3.0.CO;2-5},
  volume       = {53},
  year         = {2000},
}