Toronto Probability Seminar

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Intermediate Disorder for Directed Polymers in Dimension 1+1, and the Continuum Random Polymer

by Tom Alberts | University of Toronto
Time: 16:00  (Monday, Sep. 13, 2010)
Location: Stewart Library, Fields Institute, 222 College St.
The 1+1 dimensional directed polymer model is a Gibbs measure on simple random walk paths of a prescribed length. The weights for the measure are determined by a random environment occupying space-time lattice sites, and the measure favors paths to which the environment gives high energy. For each inverse temperature $\beta$ the polymer is said to be in the weak disorder regime if the environment has little effect on it, and the strong disorder regime otherwise. In dimension 1+1 it turns out that all positive $\beta$ are in the strong disorder regime. I will introduce a new regime called intermediate disorder, which is accessed by scaling the inverse temperature to zero with the length $n$ of the polymer. The precise scaling is $\beta n^{-1/4}$. The most interesting result is that under this scaling the polymer has diffusive fluctuations, but the fluctuations themselves are not Gaussian. Instead they are still coupled to the random environment, and their distribution is intimately related to the Tracy-Widom distribution for the largest eigenvalue of a random matrix from the GUE. More recent work also indicates that we can take a scaling limit of the entire intermediate disorder regime to construct a continuous random path under the effect of a continuum random environment. We call the scaling limit the continuum random polymer. I will discuss a few properties of the continuum random polymer and its intimate connection to the stochastic heat equation in one dimension. Joint work with Kostya Khanin and Jeremy Quastel.

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)