@article{14499, abstract = {An n-vertex graph is called C-Ramsey if it has no clique or independent set of size Clog2n (i.e., if it has near-optimal Ramsey behavior). In this paper, we study edge statistics in Ramsey graphs, in particular obtaining very precise control of the distribution of the number of edges in a random vertex subset of a C-Ramsey graph. This brings together two ongoing lines of research: the study of ‘random-like’ properties of Ramsey graphs and the study of small-ball probability for low-degree polynomials of independent random variables. The proof proceeds via an ‘additive structure’ dichotomy on the degree sequence and involves a wide range of different tools from Fourier analysis, random matrix theory, the theory of Boolean functions, probabilistic combinatorics and low-rank approximation. In particular, a key ingredient is a new sharpened version of the quadratic Carbery–Wright theorem on small-ball probability for polynomials of Gaussians, which we believe is of independent interest. One of the consequences of our result is the resolution of an old conjecture of Erdős and McKay, for which Erdős reiterated in several of his open problem collections and for which he offered one of his notorious monetary prizes.}, author = {Kwan, Matthew Alan and Sah, Ashwin and Sauermann, Lisa and Sawhney, Mehtaab}, issn = {2050-5086}, journal = {Forum of Mathematics, Pi}, keywords = {Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics, Statistics and Probability, Algebra and Number Theory, Analysis}, publisher = {Cambridge University Press}, title = {{Anticoncentration in Ramsey graphs and a proof of the Erdős–McKay conjecture}}, doi = {10.1017/fmp.2023.17}, volume = {11}, year = {2023}, } @article{14192, abstract = {For the Fröhlich model of the large polaron, we prove that the ground state energy as a function of the total momentum has a unique global minimum at momentum zero. This implies the non-existence of a ground state of the translation invariant Fröhlich Hamiltonian and thus excludes the possibility of a localization transition at finite coupling.}, author = {Lampart, Jonas and Mitrouskas, David Johannes and Mysliwy, Krzysztof}, issn = {1572-9656}, journal = {Mathematical Physics, Analysis and Geometry}, keywords = {Geometry and Topology, Mathematical Physics}, number = {3}, publisher = {Springer Nature}, title = {{On the global minimum of the energy–momentum relation for the polaron}}, doi = {10.1007/s11040-023-09460-x}, volume = {26}, year = {2023}, } @article{14756, abstract = {We prove the r-spin cobordism hypothesis in the setting of (weak) 2-categories for every positive integer r: the 2-groupoid of 2-dimensional fully extended r-spin TQFTs with given target is equivalent to the homotopy fixed points of an induced Spin 2r -action. In particular, such TQFTs are classified by fully dualisable objects together with a trivialisation of the rth power of their Serre automorphisms. For r=1, we recover the oriented case (on which our proof builds), while ordinary spin structures correspond to r=2. To construct examples, we explicitly describe Spin 2r​-homotopy fixed points in the equivariant completion of any symmetric monoidal 2-category. We also show that every object in a 2-category of Landau–Ginzburg models gives rise to fully extended spin TQFTs and that half of these do not factor through the oriented bordism 2-category.}, author = {Carqueville, Nils and Szegedy, Lorant}, issn = {1663-487X}, journal = {Quantum Topology}, keywords = {Geometry and Topology, Mathematical Physics}, number = {3}, pages = {467--532}, publisher = {European Mathematical Society}, title = {{Fully extended r-spin TQFTs}}, doi = {10.4171/qt/193}, volume = {14}, year = {2023}, } @article{10600, abstract = {We show that recent results on adiabatic theory for interacting gapped many-body systems on finite lattices remain valid in the thermodynamic limit. More precisely, we prove a generalized super-adiabatic theorem for the automorphism group describing the infinite volume dynamics on the quasi-local algebra of observables. The key assumption is the existence of a sequence of gapped finite volume Hamiltonians, which generates the same infinite volume dynamics in the thermodynamic limit. Our adiabatic theorem also holds for certain perturbations of gapped ground states that close the spectral gap (so it is also an adiabatic theorem for resonances and, in this sense, “generalized”), and it provides an adiabatic approximation to all orders in the adiabatic parameter (a property often called “super-adiabatic”). In addition to the existing results for finite lattices, we also perform a resummation of the adiabatic expansion and allow for observables that are not strictly local. Finally, as an application, we prove the validity of linear and higher order response theory for our class of perturbations for infinite systems. While we consider the result and its proof as new and interesting in itself, we also lay the foundation for the proof of an adiabatic theorem for systems with a gap only in the bulk, which will be presented in a follow-up article.}, author = {Henheik, Sven Joscha and Teufel, Stefan}, issn = {1089-7658}, journal = {Journal of Mathematical Physics}, keywords = {mathematical physics, statistical and nonlinear physics}, number = {1}, publisher = {AIP Publishing}, title = {{Adiabatic theorem in the thermodynamic limit: Systems with a uniform gap}}, doi = {10.1063/5.0051632}, volume = {63}, year = {2022}, } @article{10642, abstract = {Based on a result by Yarotsky (J Stat Phys 118, 2005), we prove that localized but otherwise arbitrary perturbations of weakly interacting quantum spin systems with uniformly gapped on-site terms change the ground state of such a system only locally, even if they close the spectral gap. We call this a strong version of the local perturbations perturb locally (LPPL) principle which is known to hold for much more general gapped systems, but only for perturbations that do not close the spectral gap of the Hamiltonian. We also extend this strong LPPL-principle to Hamiltonians that have the appropriate structure of gapped on-site terms and weak interactions only locally in some region of space. While our results are technically corollaries to a theorem of Yarotsky, we expect that the paradigm of systems with a locally gapped ground state that is completely insensitive to the form of the Hamiltonian elsewhere extends to other situations and has important physical consequences.}, author = {Henheik, Sven Joscha and Teufel, Stefan and Wessel, Tom}, issn = {1573-0530}, journal = {Letters in Mathematical Physics}, keywords = {mathematical physics, statistical and nonlinear physics}, number = {1}, publisher = {Springer Nature}, title = {{Local stability of ground states in locally gapped and weakly interacting quantum spin systems}}, doi = {10.1007/s11005-021-01494-y}, volume = {112}, year = {2022}, } @article{10643, abstract = {We prove a generalised super-adiabatic theorem for extended fermionic systems assuming a spectral gap only in the bulk. More precisely, we assume that the infinite system has a unique ground state and that the corresponding Gelfand–Naimark–Segal Hamiltonian has a spectral gap above its eigenvalue zero. Moreover, we show that a similar adiabatic theorem also holds in the bulk of finite systems up to errors that vanish faster than any inverse power of the system size, although the corresponding finite-volume Hamiltonians need not have a spectral gap. }, author = {Henheik, Sven Joscha and Teufel, Stefan}, issn = {2050-5094}, journal = {Forum of Mathematics, Sigma}, keywords = {computational mathematics, discrete mathematics and combinatorics, geometry and topology, mathematical physics, statistics and probability, algebra and number theory, theoretical computer science, analysis}, publisher = {Cambridge University Press}, title = {{Adiabatic theorem in the thermodynamic limit: Systems with a gap in the bulk}}, doi = {10.1017/fms.2021.80}, volume = {10}, year = {2022}, } @article{10623, abstract = {We investigate the BCS critical temperature Tc in the high-density limit and derive an asymptotic formula, which strongly depends on the behavior of the interaction potential V on the Fermi-surface. Our results include a rigorous confirmation for the behavior of Tc at high densities proposed by Langmann et al. (Phys Rev Lett 122:157001, 2019) and identify precise conditions under which superconducting domes arise in BCS theory.}, author = {Henheik, Sven Joscha}, issn = {1572-9656}, journal = {Mathematical Physics, Analysis and Geometry}, keywords = {geometry and topology, mathematical physics}, number = {1}, publisher = {Springer Nature}, title = {{The BCS critical temperature at high density}}, doi = {10.1007/s11040-021-09415-0}, volume = {25}, year = {2022}, } @article{11783, abstract = {We consider a gas of N bosons with interactions in the mean-field scaling regime. We review the proof of an asymptotic expansion of its low-energy spectrum, eigenstates, and dynamics, which provides corrections to Bogoliubov theory to all orders in 1/ N. This is based on joint works with Petrat, Pickl, Seiringer, and Soffer. In addition, we derive a full asymptotic expansion of the ground state one-body reduced density matrix.}, author = {Bossmann, Lea}, issn = {1089-7658}, journal = {Journal of Mathematical Physics}, keywords = {Mathematical Physics, Statistical and Nonlinear Physics}, number = {6}, publisher = {AIP Publishing}, title = {{Low-energy spectrum and dynamics of the weakly interacting Bose gas}}, doi = {10.1063/5.0089983}, volume = {63}, year = {2022}, } @article{11917, abstract = {We study the many-body dynamics of an initially factorized bosonic wave function in the mean-field regime. We prove large deviation estimates for the fluctuations around the condensate. We derive an upper bound extending a recent result to more general interactions. Furthermore, we derive a new lower bound which agrees with the upper bound in leading order.}, author = {Rademacher, Simone Anna Elvira and Seiringer, Robert}, issn = {1572-9613}, journal = {Journal of Statistical Physics}, keywords = {Mathematical Physics, Statistical and Nonlinear Physics}, publisher = {Springer Nature}, title = {{Large deviation estimates for weakly interacting bosons}}, doi = {10.1007/s10955-022-02940-4}, volume = {188}, year = {2022}, } @article{12145, abstract = {In the class of strictly convex smooth boundaries each of which has no strip around its boundary foliated by invariant curves, we prove that the Taylor coefficients of the “normalized” Mather’s β-function are invariant under C∞-conjugacies. In contrast, we prove that any two elliptic billiard maps are C0-conjugate near their respective boundaries, and C∞-conjugate, near the boundary and away from a line passing through the center of the underlying ellipse. We also prove that, if the billiard maps corresponding to two ellipses are topologically conjugate, then the two ellipses are similar.}, author = {Koudjinan, Edmond and Kaloshin, Vadim}, issn = {1468-4845}, journal = {Regular and Chaotic Dynamics}, keywords = {Mechanical Engineering, Applied Mathematics, Mathematical Physics, Modeling and Simulation, Statistical and Nonlinear Physics, Mathematics (miscellaneous)}, number = {6}, pages = {525--537}, publisher = {Springer Nature}, title = {{On some invariants of Birkhoff billiards under conjugacy}}, doi = {10.1134/S1560354722050021}, volume = {27}, year = {2022}, } @article{12148, abstract = {We prove a general local law for Wigner matrices that optimally handles observables of arbitrary rank and thus unifies the well-known averaged and isotropic local laws. As an application, we prove a central limit theorem in quantum unique ergodicity (QUE): that is, we show that the quadratic forms of a general deterministic matrix A on the bulk eigenvectors of a Wigner matrix have approximately Gaussian fluctuation. For the bulk spectrum, we thus generalise our previous result [17] as valid for test matrices A of large rank as well as the result of Benigni and Lopatto [7] as valid for specific small-rank observables.}, author = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J}, issn = {2050-5094}, journal = {Forum of Mathematics, Sigma}, keywords = {Computational Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics, Statistics and Probability, Algebra and Number Theory, Theoretical Computer Science, Analysis}, publisher = {Cambridge University Press}, title = {{Rank-uniform local law for Wigner matrices}}, doi = {10.1017/fms.2022.86}, volume = {10}, year = {2022}, } @article{12232, abstract = {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.}, author = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J}, issn = {1424-0661}, journal = {Annales Henri Poincaré}, keywords = {Mathematical Physics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics}, number = {11}, pages = {3981--4002}, publisher = {Springer Nature}, title = {{Density of small singular values of the shifted real Ginibre ensemble}}, doi = {10.1007/s00023-022-01188-8}, volume = {23}, year = {2022}, } @article{12243, abstract = {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)]. }, author = {Cipolloni, Giorgio and Erdös, László and Schröder, Dominik J and Xu, Yuanyuan}, issn = {1089-7658}, journal = {Journal of Mathematical Physics}, keywords = {Mathematical Physics, Statistical and Nonlinear Physics}, number = {10}, publisher = {AIP Publishing}, title = {{Directional extremal statistics for Ginibre eigenvalues}}, doi = {10.1063/5.0104290}, volume = {63}, year = {2022}, } @article{12259, abstract = {Theoretical foundations of chaos have been predominantly laid out for finite-dimensional dynamical systems, such as the three-body problem in classical mechanics and the Lorenz model in dissipative systems. In contrast, many real-world chaotic phenomena, e.g., weather, arise in systems with many (formally infinite) degrees of freedom, which limits direct quantitative analysis of such systems using chaos theory. In the present work, we demonstrate that the hydrodynamic pilot-wave systems offer a bridge between low- and high-dimensional chaotic phenomena by allowing for a systematic study of how the former connects to the latter. Specifically, we present experimental results, which show the formation of low-dimensional chaotic attractors upon destabilization of regular dynamics and a final transition to high-dimensional chaos via the merging of distinct chaotic regions through a crisis bifurcation. Moreover, we show that the post-crisis dynamics of the system can be rationalized as consecutive scatterings from the nonattracting chaotic sets with lifetimes following exponential distributions. }, author = {Choueiri, George H and Suri, Balachandra and Merrin, Jack and Serbyn, Maksym and Hof, Björn and Budanur, Nazmi B}, issn = {1089-7682}, journal = {Chaos: An Interdisciplinary Journal of Nonlinear Science}, keywords = {Applied Mathematics, General Physics and Astronomy, Mathematical Physics, Statistical and Nonlinear Physics}, number = {9}, publisher = {AIP Publishing}, title = {{Crises and chaotic scattering in hydrodynamic pilot-wave experiments}}, doi = {10.1063/5.0102904}, volume = {32}, year = {2022}, } @article{11732, abstract = {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.}, author = {Henheik, Sven Joscha and Lauritsen, Asbjørn Bækgaard}, issn = {1572-9613}, journal = {Journal of Statistical Physics}, keywords = {Mathematical Physics, Statistical and Nonlinear Physics}, publisher = {Springer Nature}, title = {{The BCS energy gap at high density}}, doi = {10.1007/s10955-022-02965-9}, volume = {189}, year = {2022}, } @article{12246, abstract = {The Lieb–Oxford inequality provides a lower bound on the Coulomb energy of a classical system of N identical charges only in terms of their one-particle density. We prove here a new estimate on the best constant in this inequality. Numerical evaluation provides the value 1.58, which is a significant improvement to the previously known value 1.64. The best constant has recently been shown to be larger than 1.44. In a second part, we prove that the constant can be reduced to 1.25 when the inequality is restricted to Hartree–Fock states. This is the first proof that the exchange term is always much lower than the full indirect Coulomb energy.}, author = {Lewin, Mathieu and Lieb, Elliott H. and Seiringer, Robert}, issn = {1573-0530}, journal = {Letters in Mathematical Physics}, keywords = {Mathematical Physics, Statistical and Nonlinear Physics}, number = {5}, publisher = {Springer Nature}, title = {{Improved Lieb–Oxford bound on the indirect and exchange energies}}, doi = {10.1007/s11005-022-01584-5}, volume = {112}, year = {2022}, } @article{9285, abstract = {We first review the problem of a rigorous justification of Kubo’s formula for transport coefficients in gapped extended Hamiltonian quantum systems at zero temperature. In particular, the theoretical understanding of the quantum Hall effect rests on the validity of Kubo’s formula for such systems, a connection that we review briefly as well. We then highlight an approach to linear response theory based on non-equilibrium almost-stationary states (NEASS) and on a corresponding adiabatic theorem for such systems that was recently proposed and worked out by one of us in [51] for interacting fermionic systems on finite lattices. In the second part of our paper, we show how to lift the results of [51] to infinite systems by taking a thermodynamic limit.}, author = {Henheik, Sven Joscha and Teufel, Stefan}, issn = {0129-055X}, journal = {Reviews in Mathematical Physics}, keywords = {Mathematical Physics, Statistical and Nonlinear Physics}, number = {01}, publisher = {World Scientific Publishing}, title = {{Justifying Kubo’s formula for gapped systems at zero temperature: A brief review and some new results}}, doi = {10.1142/s0129055x20600041}, volume = {33}, year = {2021}, } @article{9891, abstract = {Extending on ideas of Lewin, Lieb, and Seiringer [Phys. Rev. B 100, 035127 (2019)], we present a modified “floating crystal” trial state for jellium (also known as the classical homogeneous electron gas) with density equal to a characteristic function. This allows us to show that three definitions of the jellium energy coincide in dimensions d ≥ 2, thus extending the result of Cotar and Petrache [“Equality of the Jellium and uniform electron gas next-order asymptotic terms for Coulomb and Riesz potentials,” arXiv: 1707.07664 (2019)] and Lewin, Lieb, and Seiringer [Phys. Rev. B 100, 035127 (2019)] that the three definitions coincide in dimension d ≥ 3. We show that the jellium energy is also equivalent to a “renormalized energy” studied in a series of papers by Serfaty and others, and thus, by the work of Bétermin and Sandier [Constr. Approximation 47, 39–74 (2018)], we relate the jellium energy to the order n term in the logarithmic energy of n points on the unit 2-sphere. We improve upon known lower bounds for this renormalized energy. Additionally, we derive formulas for the jellium energy of periodic configurations.}, author = {Lauritsen, Asbjørn Bækgaard}, issn = {1089-7658}, journal = {Journal of Mathematical Physics}, keywords = {Mathematical Physics, Statistical and Nonlinear Physics}, number = {8}, publisher = {AIP Publishing}, title = {{Floating Wigner crystal and periodic jellium configurations}}, doi = {10.1063/5.0053494}, volume = {62}, year = {2021}, } @article{9973, abstract = {In this article we introduce a complete gradient estimate for symmetric quantum Markov semigroups on von Neumann algebras equipped with a normal faithful tracial state, which implies semi-convexity of the entropy with respect to the recently introduced noncommutative 2-Wasserstein distance. We show that this complete gradient estimate is stable under tensor products and free products and establish its validity for a number of examples. As an application we prove a complete modified logarithmic Sobolev inequality with optimal constant for Poisson-type semigroups on free group factors.}, author = {Wirth, Melchior and Zhang, Haonan}, issn = {1432-0916}, journal = {Communications in Mathematical Physics}, keywords = {Mathematical Physics, Statistical and Nonlinear Physics}, pages = {761–791}, publisher = {Springer Nature}, title = {{Complete gradient estimates of quantum Markov semigroups}}, doi = {10.1007/s00220-021-04199-4}, volume = {387}, year = {2021}, } @article{9121, abstract = {We show that the energy gap for the BCS gap equation is Ξ=μ(8e−2+o(1))exp(π2μ−−√a) in the low density limit μ→0. Together with the similar result for the critical temperature by Hainzl and Seiringer (Lett Math Phys 84: 99–107, 2008), this shows that, in the low density limit, the ratio of the energy gap and critical temperature is a universal constant independent of the interaction potential V. The results hold for a class of potentials with negative scattering length a and no bound states.}, author = {Lauritsen, Asbjørn Bækgaard}, issn = {1573-0530}, journal = {Letters in Mathematical Physics}, keywords = {Mathematical Physics, Statistical and Nonlinear Physics}, publisher = {Springer Nature}, title = {{The BCS energy gap at low density}}, doi = {10.1007/s11005-021-01358-5}, volume = {111}, year = {2021}, }