Toronto Probability Seminar

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The top eigenvalue of a spiked real sample covariance matrix: A continuum operator limit approach

by Alex Bloemendal | University of Toronto
Time: 16:00  (Monday, Mar. 01, 2010)
Location: FI (Stewart Library), Fields Institute, 222 College St.
Simple trends in high-dimensional data sets are modeled by spiked real Gaussian sample covariance matrices. In the large-size limit, the behaviour of the top eigenvalue exhibits a phase transition as a function of the strength of the trend. Using a stochastic operator limit approach, we show that the top eigenvalue has an asymptotic distribution near the phase transition and give several characterizations of the limit laws; one of these involves only a linear boundary value problem. In the well-studied complex case, our PDE description reproduces known explicit formulas and yields a simple new derivation of the Painleve formula for the Tracy-Widom distribution. Joint work with Balint Virag.

Dates in this series

· Monday, Jan. 11, 2010: Scaling Exponents for Directed Polymers in 1+1 Dimensions, Part II  (Tom Alberts )
· Monday, Jan. 18, 2010: Eigenvector Localization for Random Band Matrices with Power Law Band Width (Jeffrey Schenker )
· Monday, Feb. 08, 2010: The prolific backbone of a superdiffusion (Andreas Kyprianou)
· Monday, Feb. 22, 2010: Extremal laws in the real Ginibre matrix ensemble (Brian Rider)
· Monday, Mar. 01, 2010: The top eigenvalue of a spiked real sample covariance matrix: A continuum operator limit approach (Alex Bloemendal)
· Monday, Mar. 08, 2010: Random walks on the two dimensional uniform spanning tree (Robert Masson)
· Monday, Mar. 15, 2010: Disordered-polymer depinning transitions: an overview (Ken Alexander)
· Friday, Mar. 19, 2010: Nonconcentration of Return Times (Ori Gurel-Gurevich)
· Monday, Mar. 22, 2010: Random Matrices with Poisson Eigenvalue Statistics (Mihai Stoiciu)
· Monday, Mar. 29, 2010: Questions about Large Graphs of Bounded Degree (Miklos Abert)
· Monday, Apr. 05, 2010: Directed Polymers in a Random Environment with Heavy Tails (Antonio Auffinger)
· Monday, Apr. 12, 2010: Wick formula for Haar unitary matrices (Jonathan Novak)
· Monday, Apr. 26, 2010: Probability Questions in Spin Glasses (Louis-Pierre Arguin)
· Monday, Jun. 28, 2010: A combinatorial result with applications to random walks (Mark Holmes)
· Wednesday, Jun. 30, 2010: On the random phase conjecture for random Schrodinger operators (Christian Sadel)
· Monday, Jul. 05, 2010: Fluctuations for the KPZ universality class (Ivan Corwin)
· Friday, Jul. 30, 2010: Hermitian matrix model with spiked external source (Dong Wang)
· Monday, Sep. 13, 2010: Intermediate Disorder for Directed Polymers in Dimension 1+1, and the Continuum Random Polymer (Tom Alberts)
· Monday, Sep. 20, 2010: Biased motion in disorder: persistent discreteness, rational resonance, and stable limits (Alan Hammond)
· Monday, Sep. 27, 2010: Brunet-Derrida particle systems, free boundary problems, and Wiener-Hopf equations (Daniel Remenik)
· Monday, Oct. 18, 2010: Random complex zeroes: fluctuations and correlations (Misha Sodin)
· Monday, Oct. 25, 2010: Limiting Fluctuation Laws for Spectral Statistics of Random Matrices (Leonid Pastur)
· Monday, Nov. 01, 2010: Gauss Curvature of Random Metrics (Dmitry Jakobson)
· Monday, Nov. 08, 2010: Connection probabilities and RSW-type bounds for the two-dimensional FK Ising model (Pierre Nolin)
· Monday, Nov. 15, 2010: Meixner Random Matrices (Gerard Letac)
· Friday, Nov. 26, 2010: Nonlinear Stochastic Perturbations of Dynamical Systems (Leonid Koralov)
· Friday, Nov. 26, 2010: Particle Systems arising from an Anti-ferromagnetic Ising Model (Sunil Chhita)
· Wednesday, Jan. 05, 2011: Interpolation Percolation (Martin Zerner)
· Monday, Jan. 10, 2011: Crossing velocities for annealed random walks in random potentials (Elena Kosygina)
· Monday, Jan. 24, 2011: Finite rank perturbations of large random matrices (Alex Bloemendal)
· Monday, Feb. 07, 2011: Random Band Matrices (Sasha Sodin)
· Wednesday, Feb. 09, 2011: Stochastic Billiards (Francis Comets)
· Friday, Feb. 11, 2011: Busemann functions and cube root fluctuations in the generalized Hammersley process (Eric Cator)
· Monday, Feb. 28, 2011: Central Limit Theorems and Large Deviations for Variational Problems (Ivan Matic)
· Monday, Mar. 07, 2011: The environment viewed from the particle for random walks in random Dirichlet environment (Christophe Sabot)
· Monday, Mar. 14, 2011: Finite Approximation of Random Graphs (Vadim Kaimanovich)
· Monday, Mar. 21, 2011: Second-order and local Boltzmann-Gibbs principle and applications (Milton Jara)
· Friday, Mar. 25, 2011: Equations of the McKean-Vlasov type in finite volume (Lincoln Chayes)
· Friday, Mar. 25, 2011: Stationary Processes, Symplectic Maps, and Arnold's Conjecture (Fraydoun Rezakhanlou)
· Monday, Mar. 28, 2011: Asymmetric directed polymers in a random environment (Gregorio Moreno Flores)
· Monday, Apr. 11, 2011: Ginzburg Landau functional with external random field: minimizers and interfaces (Enza Orlandi)
· Tuesday, Apr. 26, 2011: Random transformations and related random walks in random environment on a strip (Ilya Goldsheild)
· Monday, May. 02, 2011: Fluctuations of large random matrices and second-order freeness (Emily Redelmeier)
· Monday, May. 09, 2011: An equilibrium problem for the two matrix model with a quartic potential (Maurice Duits)
· Friday, May. 13, 2011: An L space with non-Lindelof square (Peng Yinhe)
· Monday, May. 16, 2011: Exponential Functional of Lévy Processes (Mladen Savov)
· Monday, May. 30, 2011: Random walks in degenerate random environments (Mark Holmes)
· Monday, Jun. 06, 2011: On Convergence Rates to SLE for Random Lattice Curves (Fredrik Viklund)
· Monday, Sep. 12, 2011: Graph limits and the corresponding spectral theory (Balazs Szegedy)
· Friday, Oct. 14, 2011: Scaling Limits for some PDEs with Random Initial Data (Narn-Rueih Shieh)
· Friday, Oct. 21, 2011: Finite graphs and Kesten's theorem (Balint Virag)
· Friday, Oct. 28, 2011: A Stochastic Version of Multiplicative Cascades (Ben Rifkind)
· Friday, Nov. 04, 2011: Optimal transportation, extremal doubly stochastic matrices, and differential topology (Robert McCann)
· Friday, Nov. 11, 2011: Large deviations for random matrices via potential theory (Thomas Bloom)
· Wednesday, Nov. 16, 2011: Variational problems for Airy processes (Jeremy Quastel)
· Friday, Nov. 18, 2011: A unitary extension of virtual permutations (Joseph Najnudel)
· Friday, Nov. 25, 2011: Uniqueness and nonuniqueness for some stochastic PDE (Carl Mueller)
· Friday, Dec. 02, 2011: Reverse hypercontractivity and its applications  (Arnab Sen)
· Tuesday, Dec. 13, 2011: Pants decomposition of random surfaces (Robert Young)
· Friday, Dec. 16, 2011: A multi-channel 1D random Dirac operator with purely absolutely continuous spectrum (Christian Sadel)
· Monday, Jan. 16, 2012: Random sequences of symmetrizations (Almut Burchard)
· Monday, Jan. 23, 2012: Self-organized criticality in a discrete model of coagulation (Raoul Normand)
· Monday, Jan. 30, 2012: Nonintersecting processes on the half-line and discrete orthogonal polynomials (Karl Liechty)
· Monday, Feb. 06, 2012: Variational formulas for the Airy2 process (Daniel Remenik)
· Monday, Feb. 13, 2012: Localization and delocalization of eigenvectors for heavy-tailed random matrices (Charles Bordenave)
· Monday, Feb. 27, 2012: Randomly forced Burgers equation in noncompact setting (Yuri Bakhtin)
· Monday, Mar. 05, 2012: Normal approximation for a random elliptic PDE (James Nolen)
· Wednesday, Mar. 14, 2012: The effect of variable drift on Brownian motion and the Wiener sausage (Perla Sousi)
· Monday, Apr. 02, 2012: Scaling exponents of lattice gases (Benedek Valko)
· Monday, Apr. 16, 2012: Product-form invariant measures for Brownian motion with drift satisfying a skew-symmetry type condition (Janosch Ortmann)
· Monday, May. 07, 2012: Criteria for ballistic behavior of random walks in random environment (Alejandro F. Ramirez)
· Monday, Oct. 01, 2012: Independent sets and the minimum eigenvalue in transitive graphs (Viktor Harangi)
· Monday, Nov. 12, 2012: How far does stuff get in an interacting system of isometric random walks on Z^d? (Jeremy Quastel)
· Monday, Nov. 26, 2012: Random groups and property (T) (Michal Kotowski)
· Monday, Dec. 03, 2012: Determinantal line ensembles (Daniel Remenik)
· Thursday, Dec. 13, 2012: Half planar maps (Omer Angel)