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 - TY - JOUR AB - We study the BCS energy gap Ξ in the high–density limit and derive an asymptotic formula, which strongly depends on the strength of the interaction potential V on the Fermi surface. In combination with the recent result by one of us (Math. Phys. Anal. Geom. 25, 3, 2022) on the critical temperature Tc at high densities, we prove the universality of the ratio of the energy gap and the critical temperature. AU - Henheik, Sven Joscha AU - Lauritsen, Asbjørn Bækgaard ID - 11732 JF - Journal of Statistical Physics KW - Mathematical Physics KW - Statistical and Nonlinear Physics SN - 0022-4715 TI - The BCS energy gap at high density VL - 189 ER - TY - JOUR AB - We study the overlaps between right and left eigenvectors for random matrices of the spherical ensemble, as well as truncated unitary ensembles in the regime where half of the matrix at least is truncated. These two integrable models exhibit a form of duality, and the essential steps of our investigation can therefore be performed in parallel. In every case, conditionally on all eigenvalues, diagonal overlaps are shown to be distributed as a product of independent random variables with explicit distributions. This enables us to prove that the scaled diagonal overlaps, conditionally on one eigenvalue, converge in distribution to a heavy-tail limit, namely, the inverse of a γ2 distribution. We also provide formulae for the conditional expectation of diagonal and off-diagonal overlaps, either with respect to one eigenvalue, or with respect to the whole spectrum. These results, analogous to what is known for the complex Ginibre ensemble, can be obtained in these cases thanks to integration techniques inspired from a previous work by Forrester & Krishnapur. AU - Dubach, Guillaume ID - 10285 JF - Electronic Journal of Probability TI - On eigenvector statistics in the spherical and truncated unitary ensembles VL - 26 ER - TY - GEN AB - We consider a model of the Riemann zeta function on the critical axis and study its maximum over intervals of length (log T)θ, where θ is either fixed or tends to zero at a suitable rate. It is shown that the deterministic level of the maximum interpolates smoothly between the ones of log-correlated variables and of i.i.d. random variables, exhibiting a smooth transition ‘from 3/4 to 1/4’ in the second order. This provides a natural context where extreme value statistics of log-correlated variables with time-dependent variance and rate occur. A key ingredient of the proof is a precise upper tail tightness estimate for the maximum of the model on intervals of size one, that includes a Gaussian correction. This correction is expected to be present for the Riemann zeta function and pertains to the question of the correct order of the maximum of the zeta function in large intervals. AU - Arguin, Louis-Pierre AU - Dubach, Guillaume AU - Hartung, Lisa ID - 9230 T2 - arXiv TI - Maxima of a random model of the Riemann zeta function over intervals of varying length ER - TY - GEN AB - We comment on two formal proofs of Fermat's sum of two squares theorem, written using the Mathematical Components libraries of the Coq proof assistant. The first one follows Zagier's celebrated one-sentence proof; the second follows David Christopher's recent new proof relying on partition-theoretic arguments. Both formal proofs rely on a general property of involutions of finite sets, of independent interest. The proof technique consists for the most part of automating recurrent tasks (such as case distinctions and computations on natural numbers) via ad hoc tactics. AU - Dubach, Guillaume AU - Mühlböck, Fabian ID - 9281 T2 - arXiv TI - Formal verification of Zagier's one-sentence proof ER - TY - JOUR AB - It is well known that special Kubo-Ando operator means admit divergence center interpretations, moreover, they are also mean squared error estimators for certain metrics on positive definite operators. In this paper we give a divergence center interpretation for every symmetric Kubo-Ando mean. This characterization of the symmetric means naturally leads to a definition of weighted and multivariate versions of a large class of symmetric Kubo-Ando means. We study elementary properties of these weighted multivariate means, and note in particular that in the special case of the geometric mean we recover the weighted A#H-mean introduced by Kim, Lawson, and Lim. AU - Pitrik, József AU - Virosztek, Daniel ID - 8373 JF - Linear Algebra and its Applications KW - Kubo-Ando mean KW - weighted multivariate mean KW - barycenter SN - 0024-3795 TI - A divergence center interpretation of general symmetric Kubo-Ando means, and related weighted multivariate operator means VL - 609 ER -