@article{8601,
  abstract     = {We consider large non-Hermitian real or complex random matrices X with independent, identically distributed centred entries. We prove that their local eigenvalue statistics near the spectral edge, the unit circle, coincide with those of the Ginibre ensemble, i.e. when the matrix elements of X are Gaussian. This result is the non-Hermitian counterpart of the universality of the Tracy–Widom distribution at the spectral edges of the Wigner ensemble.},
  author       = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J},
  issn         = {14322064},
  journal      = {Probability Theory and Related Fields},
  publisher    = {Springer Nature},
  title        = {{Edge universality for non-Hermitian random matrices}},
  doi          = {10.1007/s00440-020-01003-7},
  year         = {2021},
}

@article{319,
  abstract     = {We study spaces of modelled distributions with singular behaviour near the boundary of a domain that, in the context of the theory of regularity structures, allow one to give robust solution theories for singular stochastic PDEs with boundary conditions. The calculus of modelled distributions established in Hairer (Invent Math 198(2):269–504, 2014. https://doi.org/10.1007/s00222-014-0505-4) is extended to this setting. We formulate and solve fixed point problems in these spaces with a class of kernels that is sufficiently large to cover in particular the Dirichlet and Neumann heat kernels. These results are then used to provide solution theories for the KPZ equation with Dirichlet and Neumann boundary conditions and for the 2D generalised parabolic Anderson model with Dirichlet boundary conditions. In the case of the KPZ equation with Neumann boundary conditions, we show that, depending on the class of mollifiers one considers, a “boundary renormalisation” takes place. In other words, there are situations in which a certain boundary condition is applied to an approximation to the KPZ equation, but the limiting process is the Hopf–Cole solution to the KPZ equation with a different boundary condition.},
  author       = {Gerencser, Mate and Hairer, Martin},
  issn         = {14322064},
  journal      = {Probability Theory and Related Fields},
  number       = {3-4},
  pages        = {697–758},
  publisher    = {Springer},
  title        = {{Singular SPDEs in domains with boundaries}},
  doi          = {10.1007/s00440-018-0841-1},
  volume       = {173},
  year         = {2019},
}

@article{429,
  abstract     = {We consider real symmetric or complex hermitian random matrices with correlated entries. We prove local laws for the resolvent and universality of the local eigenvalue statistics in the bulk of the spectrum. The correlations have fast decay but are otherwise of general form. The key novelty is the detailed stability analysis of the corresponding matrix valued Dyson equation whose solution is the deterministic limit of the resolvent.},
  author       = {Ajanki, Oskari H and Erdös, László and Krüger, Torben H},
  issn         = {14322064},
  journal      = {Probability Theory and Related Fields},
  number       = {1-2},
  pages        = {293–373},
  publisher    = {Springer},
  title        = {{Stability of the matrix Dyson equation and random matrices with correlations}},
  doi          = {10.1007/s00440-018-0835-z},
  volume       = {173},
  year         = {2019},
}

@article{1528,
  abstract     = {We consider N×N Hermitian random matrices H consisting of blocks of size M≥N6/7. The matrix elements are i.i.d. within the blocks, close to a Gaussian in the four moment matching sense, but their distribution varies from block to block to form a block-band structure, with an essential band width M. We show that the entries of the Green’s function G(z)=(H−z)−1 satisfy the local semicircle law with spectral parameter z=E+iη down to the real axis for any η≫N−1, using a combination of the supersymmetry method inspired by Shcherbina (J Stat Phys 155(3): 466–499, 2014) and the Green’s function comparison strategy. Previous estimates were valid only for η≫M−1. The new estimate also implies that the eigenvectors in the middle of the spectrum are fully delocalized.},
  author       = {Bao, Zhigang and Erdös, László},
  issn         = {01788051},
  journal      = {Probability Theory and Related Fields},
  number       = {3-4},
  pages        = {673 -- 776},
  publisher    = {Springer},
  title        = {{Delocalization for a class of random block band matrices}},
  doi          = {10.1007/s00440-015-0692-y},
  volume       = {167},
  year         = {2017},
}

@article{1337,
  abstract     = {We consider the local eigenvalue distribution of large self-adjoint N×N random matrices H=H∗ with centered independent entries. In contrast to previous works the matrix of variances sij=\mathbbmE|hij|2 is not assumed to be stochastic. Hence the density of states is not the Wigner semicircle law. Its possible shapes are described in the companion paper (Ajanki et al. in Quadratic Vector Equations on the Complex Upper Half Plane. arXiv:1506.05095). We show that as N grows, the resolvent, G(z)=(H−z)−1, converges to a diagonal matrix, diag(m(z)), where m(z)=(m1(z),…,mN(z)) solves the vector equation −1/mi(z)=z+∑jsijmj(z) that has been analyzed in Ajanki et al. (Quadratic Vector Equations on the Complex Upper Half Plane. arXiv:1506.05095). We prove a local law down to the smallest spectral resolution scale, and bulk universality for both real symmetric and complex hermitian symmetry classes.},
  author       = {Ajanki, Oskari H and Erdös, László and Krüger, Torben H},
  issn         = {01788051},
  journal      = {Probability Theory and Related Fields},
  number       = {3-4},
  pages        = {667 -- 727},
  publisher    = {Springer},
  title        = {{Universality for general Wigner-type matrices}},
  doi          = {10.1007/s00440-016-0740-2},
  volume       = {169},
  year         = {2017},
}