TY - JOUR AB - We review recent results on adiabatic theory for ground states of extended gapped fermionic lattice systems under several different assumptions. More precisely, we present generalized super-adiabatic theorems for extended but finite and infinite systems, assuming either a uniform gap or a gap in the bulk above the unperturbed ground state. The goal of this Review is to provide an overview of these adiabatic theorems and briefly outline the main ideas and techniques required in their proofs. AU - Henheik, Sven Joscha AU - Wessel, Tom ID - 12184 IS - 12 JF - Journal of Mathematical Physics SN - 0022-2488 TI - On adiabatic theory for extended fermionic lattice systems VL - 63 ER - TY - JOUR AB - Motivated by Kloeckner’s result on the isometry group of the quadratic Wasserstein space W2(Rn), we describe the isometry group Isom(Wp(E)) for all parameters 0 < p < ∞ and for all separable real Hilbert spaces E. In particular, we show that Wp(X) is isometrically rigid for all Polish space X whenever 0 < p < 1. This is a consequence of our more general result: we prove that W1(X) is isometrically rigid if X is a complete separable metric space that satisfies the strict triangle inequality. Furthermore, we show that this latter rigidity result does not generalise to parameters p > 1, by solving Kloeckner’s problem affirmatively on the existence of mass-splitting isometries. AU - Gehér, György Pál AU - Titkos, Tamás AU - Virosztek, Daniel ID - 12214 IS - 4 JF - Journal of the London Mathematical Society KW - General Mathematics SN - 0024-6107 TI - The isometry group of Wasserstein spaces: The Hilbertian case VL - 106 ER - TY - JOUR AB - We derive a precise asymptotic formula for the density of the small singular values of the real Ginibre matrix ensemble shifted by a complex parameter z as the dimension tends to infinity. For z away from the real axis the formula coincides with that for the complex Ginibre ensemble we derived earlier in Cipolloni et al. (Prob Math Phys 1:101–146, 2020). On the level of the one-point function of the low lying singular values we thus confirm the transition from real to complex Ginibre ensembles as the shift parameter z becomes genuinely complex; the analogous phenomenon has been well known for eigenvalues. We use the superbosonization formula (Littelmann et al. in Comm Math Phys 283:343–395, 2008) in a regime where the main contribution comes from a three dimensional saddle manifold. AU - Cipolloni, Giorgio AU - Erdös, László AU - Schröder, Dominik J ID - 12232 IS - 11 JF - Annales Henri Poincaré KW - Mathematical Physics KW - Nuclear and High Energy Physics KW - Statistical and Nonlinear Physics SN - 1424-0637 TI - Density of small singular values of the shifted real Ginibre ensemble VL - 23 ER - TY - JOUR AB - We consider the eigenvalues of a large dimensional real or complex Ginibre matrix in the region of the complex plane where their real parts reach their maximum value. This maximum follows the Gumbel distribution and that these extreme eigenvalues form a Poisson point process as the dimension asymptotically tends to infinity. In the complex case, these facts have already been established by Bender [Probab. Theory Relat. Fields 147, 241 (2010)] and in the real case by Akemann and Phillips [J. Stat. Phys. 155, 421 (2014)] even for the more general elliptic ensemble with a sophisticated saddle point analysis. The purpose of this article is to give a very short direct proof in the Ginibre case with an effective error term. Moreover, our estimates on the correlation kernel in this regime serve as a key input for accurately locating [Formula: see text] for any large matrix X with i.i.d. entries in the companion paper [G. Cipolloni et al., arXiv:2206.04448 (2022)]. AU - Cipolloni, Giorgio AU - Erdös, László AU - Schröder, Dominik J AU - Xu, Yuanyuan ID - 12243 IS - 10 JF - Journal of Mathematical Physics KW - Mathematical Physics KW - Statistical and Nonlinear Physics SN - 0022-2488 TI - Directional extremal statistics for Ginibre eigenvalues VL - 63 ER - TY - JOUR AB - We prove local laws, i.e. optimal concentration estimates for arbitrary products of resolvents of a Wigner random matrix with deterministic matrices in between. We find that the size of such products heavily depends on whether some of the deterministic matrices are traceless. Our estimates correctly account for this dependence and they hold optimally down to the smallest possible spectral scale. AU - Cipolloni, Giorgio AU - Erdös, László AU - Schröder, Dominik J ID - 12290 JF - Electronic Journal of Probability KW - Statistics KW - Probability and Uncertainty KW - Statistics and Probability TI - Optimal multi-resolvent local laws for Wigner matrices VL - 27 ER -