Antti Knowles

Contact information

Email: [firstname].[lastname]@unige.ch
Phone: +41 22 379 11 66
Office: 612
Address: University of Geneva, Section of Mathematics, Rue du Conseil-Général 7-9, 1205 Genève, Switzerland

Interests

My research is at the interface of probability, analysis, and mathematical physics. I work mainly on random matrices, random graphs, statistical mechanics, stochastic processes, high-dimensional statistics, quantum field theory, and quantum dynamics.

Research

• The Euclidean φ⁴₂ theory as a limit of an interacting Bose gas
  J. Fröhlich, A. Knowles, B. Schlein, and V. Sohinger
  Preprint arXiv:2201.07632, 62 pages, to appear in J. Eur. Math. Soc.
• The completely delocalized region of the Erdős-Rényi graph
  J. Alt, R. Ducatez, A. Knowles
  Elect. Comm. Prob. 27 (2022), 1-9.
• Poisson statistics and localization at the spectral edge of sparse Erdős-Rényi graphs
  J. Alt, R. Ducatez, A. Knowles
  Preprint arXiv:2106.12519, 79 pages, to appear in Ann. Prob.
• Interacting loop ensembles and Bose gases
  J. Fröhlich, A. Knowles, B. Schlein, and V. Sohinger
  Preprint arXiv:2012.05110, 55 pages, to appear in Ann. Henri Poincaré.
• Delocalization transition for critical Erdős-Rényi graphs
  J. Alt, R. Ducatez, A. Knowles
  Comm. Math. Phys. 388 (2021), 507-579.
• Fluctuations of extreme eigenvalues of sparse Erdős-Rényi graphs
  Y. He, A. Knowles
  Prob. Theor. Rel. Fields. 180 (2021), 985-1056.
• A path-integral analysis of interacting Bose gases and loop gases
  J. Fröhlich, A. Knowles, B. Schlein, and V. Sohinger
  J. Stat. Phys. (Special issue in honour of Joel Lebowitz) 180 (2020), 810-831.
• The mean-field limit of quantum Bose gases at positive temperature
  J. Fröhlich, A. Knowles, B. Schlein, and V. Sohinger
  Preprint arXiv:2001.01546, 74 pages, to appear in J. Amer. Math. Soc.
• Edge rigidity and universality of random regular graphs of intermediate degree
  R. Bauerschmidt, J. Huang, A. Knowles, H.-T. Yau
  Geom. Funct. Anal. 30 (2020), 693-769.
• Extremal eigenvalues of critical Erdős-Rényi graphs
  J. Alt, R. Ducatez, A. Knowles
  Ann. Prob. 49 (2021), 1347-1401.
• Mesoscopic eigenvalue density correlations of Wigner matrices
  Y. He, A. Knowles
  Prob. Theor. Rel. Fields 177 (2020), 147-216.
• Local law and complete eigenvector delocalization for supercritical Erdős-Rényi graphs
  Y. He, A. Knowles, M. Marcozzi
  Ann. Prob. 47 (2019), 3278-3302.
• Largest eigenvalues of sparse inhomogeneous Erdős-Rényi graphs
  F. Benaych-Georges, C. Bordenave, A. Knowles
  Ann. Prob. 47 (2019), 1653-1676.
• Spectral radii of sparse random matrices
  F. Benaych-Georges, C. Bordenave, A. Knowles
  Ann. Inst. Henri Poincaré 56 (2020), 2141-2161.
• A microscopic derivation of time-dependent correlation functions of the 1D cubic nonlinear Schrödinger equation
  J. Fröhlich, A. Knowles, B. Schlein, and V. Sohinger
  Adv. Math. 353 (2019), 67-115.
• Isotropic self-consistent equations for mean-field random matrices
  Y. He, A. Knowles, and R. Rosenthal
  Prob. Theor. Rel. Fields 171 (2017), 203-249.
• Gibbs measures of nonlinear Schrödinger equations as limits of many-body quantum states in dimensions d ≤ 3
  J. Fröhlich, A. Knowles, B. Schlein, and V. Sohinger
  Comm. Math. Phys. 356 (2017), 883-980.
• Mesoscopic eigenvalue statistics of Wigner matrices
  Y. He and A. Knowles
  Ann. Appl. Prob. 27 (2017), 1510-1550.
• Eigenvalue confinement and spectral gap for random simplicial complexes
  A. Knowles and R. Rosenthal
  Rand. Struct. Algor. 51 (2017), 506-537.
• Bulk eigenvalue statistics for random regular graphs
  R. Bauerschmidt, A. Knowles, J. Huang, and H.-T. Yau
  Ann. Prob. 45 (2017), 3626-3663.
• Local semicircle law for random regular graphs
  R. Bauerschmidt, A. Knowles, and H.-T. Yau
  Comm. Pure Appl. Math. 70 (2017), 1898-1960.
• Anisotropic local laws for random matrices
  A. Knowles and J. Yin
  Prob. Theor. Rel. Fields 169 (2017), 257-352.
• On the principal components of sample covariance matrices
  A. Bloemendal, A. Knowles, H.-T. Yau, and J. Yin
  Prob. Theor. Rel. Fields 164 (2016), 459-552.
• The Altshuler-Shklovskii formulas for random band matrices II: the general case
  L. Erdős and A. Knowles
  Ann. H. Poincaré 16 (2015), 709-799.
• The Altshuler-Shklovskii formulas for random band matrices I: the unimodular case
  L. Erdős and A. Knowles
  Comm. Math. Phys. 333 (2015), 1365-1416.
• Isotropic local laws for sample covariance and generalized Wigner matrices
  A. Bloemendal, L. Erdős, A. Knowles, H.-T. Yau, and J. Yin
  Elect. J. Prob. 19 (2014), no. 33, 54 pages.
• The local semicircle law for a general class of random matrices
  L. Erdős, A. Knowles, H.-T. Yau, and J. Yin
  Elect. J. Prob. 18 (2013), no. 59, 58 pages.
• The outliers of a deformed Wigner matrix
  A. Knowles and J. Yin
  Ann. Prob. 42 (2014), 1980-2031.
• Delocalization and diffusion profile for random band matrices
  L. Erdős, A. Knowles, H.-T. Yau, and J. Yin
  Comm. Math. Phys. 323 (2013), 367-416.
• Averaging fluctuations in resolvents of random band matrices
  L. Erdős, A. Knowles, and H.-T. Yau
  Ann. H. Poincaré 14 (2013), 1837-1925.
• The isotropic semicircle law and deformation of Wigner matrices
  A. Knowles and J. Yin
  Comm. Pure Appl. Math. 66 (2013), 1663-1750.
• Spectral statistics of Erdős-Rényi graphs II: eigenvalue spacing and the extreme eigenvalues
  L. Erdős, A. Knowles, H.-T. Yau, and J. Yin
  Comm. Math. Phys. 314 (2012), 587-640.
• Spectral statistics of Erdős-Rényi graphs I: local semicircle law
  L. Erdős, A. Knowles, H.-T. Yau, and J. Yin
  Ann. Prob. 41 (2013), 2279-2375.
• Eigenvector distribution of Wigner matrices
  A. Knowles and J. Yin
  Prob. Theor. Rel. Fields 155 (2013), 543-582.
• Quantum diffusion and delocalization for band matrices general distribution
  L. Erdős and A. Knowles
  Ann. H. Poincaré 12 (2011), 1227-1319.
• Quantum diffusion and eigenfunction delocalization in a random band matrix model
  L. Erdős and A. Knowles
  Comm. Math. Phys. 303 (2011), 509-554.
• Mean-field dynamics: singular potentials and rate of convergence
  A. Knowles and P. Pickl
  Comm. Math. Phys. 298 (2010), 101-139.
• A microscopic derivation of the time-dependent Hartree-Fock equation Coulomb two-body interaction
  J. Fröhlich and A. Knowles
  J. Stat. Phys. 145 (2011), 23-50.
• On the mean-field limit of bosons Coulomb two-body interaction
  J. Fröhlich, A. Knowles, and S. Schwarz
  Comm. Math. Phys. 288 (2009), 1023-1059.
• Semi-classical dynamics in quantum spin systems
  J. Fröhlich, A. Knowles, and E. Lenzmann
  Lett. Math. Phys. 82 (2007), no. 2-3, 275-296.
• Atomism and quantization
  J. Fröhlich, A. Knowles, and A. Pizzo
  J. Phys. A 40 (2007), 3033-3045.

Expository texts and theses

• Lectures on the local semicircle law for Wigner matrices, with F. Benaych-Georges. In Advanced Topics in Random Matrices, Panoramas et Synthèses 53 (2016), Société Mathématique de France.
• Quantum diffusion. Oberwolfach snapshot (2015).
• Limiting dynamics in large quantum systems. Doctoral Thesis (2009). ETHZ e-collection 18517.
• Lattice Yang-Mills theory and the confinement problem. Diploma (Master) Thesis at ETHZ (2005).

Past events

Conference Random Matrices and Random Landscapes in honour of Yan Fyodorov's 60th birthday.

Other

Here is my CV.

I am on the editorial boards of Annales de l’Institut Fourier, Annals of Applied Probability, L'Enseignement Mathématique, and Journal of Statistical Physics.

I am grateful to the European Research Council and the Swiss State Secretariat for Education, Research and Innovation for their support through the grants RandMat (2017-2022) and ProbQuant (2022-2027).