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 - TY - JOUR AB - In this short note, we prove that the square root of the quantum Jensen-Shannon divergence is a true metric on the cone of positive matrices, and hence in particular on the quantum state space. AU - Virosztek, Daniel ID - 9036 IS - 3 JF - Advances in Mathematics KW - General Mathematics SN - 0001-8708 TI - The metric property of the quantum Jensen-Shannon divergence VL - 380 ER - TY - JOUR AB - We extend our recent result [22] on the central limit theorem for the linear eigenvalue statistics of non-Hermitian matrices X with independent, identically distributed complex entries to the real symmetry class. We find that the expectation and variance substantially differ from their complex counterparts, reflecting (i) the special spectral symmetry of real matrices onto the real axis; and (ii) the fact that real i.i.d. matrices have many real eigenvalues. Our result generalizes the previously known special cases where either the test function is analytic [49] or the first four moments of the matrix elements match the real Gaussian [59, 44]. The key element of the proof is the analysis of several weakly dependent Dyson Brownian motions (DBMs). The conceptual novelty of the real case compared with [22] is that the correlation structure of the stochastic differentials in each individual DBM is non-trivial, potentially even jeopardising its well-posedness. AU - Cipolloni, Giorgio AU - Erdös, László AU - Schröder, Dominik J ID - 9412 JF - Electronic Journal of Probability TI - Fluctuation around the circular law for random matrices with real entries VL - 26 ER - TY - JOUR AB - We prove that the energy of any eigenvector of a sum of several independent large Wigner matrices is equally distributed among these matrices with very high precision. This shows a particularly strong microcanonical form of the equipartition principle for quantum systems whose components are modelled by Wigner matrices. AU - Bao, Zhigang AU - Erdös, László AU - Schnelli, Kevin ID - 9550 JF - Forum of Mathematics, Sigma TI - Equipartition principle for Wigner matrices VL - 9 ER - TY - JOUR AB - In the customary random matrix model for transport in quantum dots with M internal degrees of freedom coupled to a chaotic environment via 𝑁≪𝑀 channels, the density 𝜌 of transmission eigenvalues is computed from a specific invariant ensemble for which explicit formula for the joint probability density of all eigenvalues is available. We revisit this problem in the large N regime allowing for (i) arbitrary ratio 𝜙:=𝑁/𝑀≤1; and (ii) general distributions for the matrix elements of the Hamiltonian of the quantum dot. In the limit 𝜙→0, we recover the formula for the density 𝜌 that Beenakker (Rev Mod Phys 69:731–808, 1997) has derived for a special matrix ensemble. We also prove that the inverse square root singularity of the density at zero and full transmission in Beenakker’s formula persists for any 𝜙<1 but in the borderline case 𝜙=1 an anomalous 𝜆−2/3 singularity arises at zero. To access this level of generality, we develop the theory of global and local laws on the spectral density of a large class of noncommutative rational expressions in large random matrices with i.i.d. entries. AU - Erdös, László AU - Krüger, Torben H AU - Nemish, Yuriy ID - 9912 JF - Annales Henri Poincaré SN - 1424-0637 TI - Scattering in quantum dots via noncommutative rational functions VL - 22 ER - TY - JOUR AB - We prove that any deterministic matrix is approximately the identity in the eigenbasis of a large random Wigner matrix with very high probability and with an optimal error inversely proportional to the square root of the dimension. Our theorem thus rigorously verifies the Eigenstate Thermalisation Hypothesis by Deutsch (Phys Rev A 43:2046–2049, 1991) for the simplest chaotic quantum system, the Wigner ensemble. In mathematical terms, we prove the strong form of Quantum Unique Ergodicity (QUE) with an optimal convergence rate for all eigenvectors simultaneously, generalizing previous probabilistic QUE results in Bourgade and Yau (Commun Math Phys 350:231–278, 2017) and Bourgade et al. (Commun Pure Appl Math 73:1526–1596, 2020). AU - Cipolloni, Giorgio AU - Erdös, László AU - Schröder, Dominik J ID - 10221 IS - 2 JF - Communications in Mathematical Physics SN - 0010-3616 TI - Eigenstate thermalization hypothesis for Wigner matrices VL - 388 ER -