Introduction Results Idea of Proof Summary Directed Polymers in Random Environment with Heavy Tails A. Auffinger O. Louidor Courant (New York University) Maryland Probability Seminar, 11/16/2009
Introduction Results Idea of Proof Summary Outline Introduction Results Idea of Proof Summary
Introduction Results Idea of Proof Summary Outline Introduction Results Idea of Proof Summary
Introduction Results Idea of Proof Summary The Model • Model for the interaction between a polymer chain and microscopic impurities in the media where it resides. • Environment σ - a random signed measure σ supported on Z + × Z d . • Example: σ ( { x , y ) } ) ; ( x , y ) ∈ Z + × Z d are i.i.d. • Polymer s - a nearest neighbor path (bridge): s : [ 0 , n ] ∩ Z → Z d , s ( 0 ) = s ( n ) = 0 , s ( x + 1 ) − s ( x ) ∈ {− 1 , + 1 } . • Interaction - an environment-dependent measure on polymer paths:
Introduction Results Idea of Proof Summary The Model • Model for the interaction between a polymer chain and microscopic impurities in the media where it resides. • Environment σ - a random signed measure σ supported on Z + × Z d . • Example: σ ( { x , y ) } ) ; ( x , y ) ∈ Z + × Z d are i.i.d. • Polymer s - a nearest neighbor path (bridge): s : [ 0 , n ] ∩ Z → Z d , s ( 0 ) = s ( n ) = 0 , s ( x + 1 ) − s ( x ) ∈ {− 1 , + 1 } . • Interaction - an environment-dependent measure on polymer paths:
Introduction Results Idea of Proof Summary The Model • Model for the interaction between a polymer chain and microscopic impurities in the media where it resides. • Environment σ - a random signed measure σ supported on Z + × Z d . • Example: σ ( { x , y ) } ) ; ( x , y ) ∈ Z + × Z d are i.i.d. • Polymer s - a nearest neighbor path (bridge): s : [ 0 , n ] ∩ Z → Z d , s ( 0 ) = s ( n ) = 0 , s ( x + 1 ) − s ( x ) ∈ {− 1 , + 1 } . • Interaction - an environment-dependent measure on polymer paths:
Introduction Results Idea of Proof Summary The Model • Model for the interaction between a polymer chain and microscopic impurities in the media where it resides. • Environment σ - a random signed measure σ supported on Z + × Z d . • Example: σ ( { x , y ) } ) ; ( x , y ) ∈ Z + × Z d are i.i.d. • Polymer s - a nearest neighbor path (bridge): s : [ 0 , n ] ∩ Z → Z d , s ( 0 ) = s ( n ) = 0 , s ( x + 1 ) − s ( x ) ∈ {− 1 , + 1 } . • Interaction - an environment-dependent measure on polymer paths:
Introduction Results Idea of Proof Summary The Model - cont’d • Explicitly: 1 µ σ exp ( − β H σ ( s )) H σ ( s ) = − σ ( s ) n ,β ( s ) = ; Q σ n ,β • β � 0 - inverse temperature, strength of interaction. • H σ - random Hamiltonian. • σ ( s ) = σ ( graph ( s )) . • Known as Directed Polymers in Random Environment (DPRE) in d + 1 dimensions. • Introduced by Huse-Henley (85).
Introduction Results Idea of Proof Summary The Model - cont’d • Explicitly: 1 µ σ exp ( − β H σ ( s )) H σ ( s ) = − σ ( s ) n ,β ( s ) = ; Q σ n ,β • β � 0 - inverse temperature, strength of interaction. • H σ - random Hamiltonian. • σ ( s ) = σ ( graph ( s )) . • Known as Directed Polymers in Random Environment (DPRE) in d + 1 dimensions. • Introduced by Huse-Henley (85).
Introduction Results Idea of Proof Summary The Model - cont’d • Explicitly: 1 µ σ exp ( − β H σ ( s )) H σ ( s ) = − σ ( s ) n ,β ( s ) = ; Q σ n ,β • β � 0 - inverse temperature, strength of interaction. • H σ - random Hamiltonian. • σ ( s ) = σ ( graph ( s )) . • Known as Directed Polymers in Random Environment (DPRE) in d + 1 dimensions. • Introduced by Huse-Henley (85).
Introduction Results Idea of Proof Summary The Questions • For β > 0 does s ( · ) behaves macroscopically differently than a Simple Random Walk bridge ( β = 0)? • Order of fluctuations. • e.g. µ σ n ,β ( � s ( n / 2 ) � 2 ) . • Pinning to certain regions in the environment. • Entropy vs. Energy. • Quenched vs. Annealed Results.
Introduction Results Idea of Proof Summary The Questions • For β > 0 does s ( · ) behaves macroscopically differently than a Simple Random Walk bridge ( β = 0)? • Order of fluctuations. • e.g. µ σ n ,β ( � s ( n / 2 ) � 2 ) . • Pinning to certain regions in the environment. • Entropy vs. Energy. • Quenched vs. Annealed Results.
Introduction Results Idea of Proof Summary The Questions • For β > 0 does s ( · ) behaves macroscopically differently than a Simple Random Walk bridge ( β = 0)? • Order of fluctuations. • e.g. µ σ n ,β ( � s ( n / 2 ) � 2 ) . • Pinning to certain regions in the environment. • Entropy vs. Energy. • Quenched vs. Annealed Results.
Introduction Results Idea of Proof Summary The Questions • For β > 0 does s ( · ) behaves macroscopically differently than a Simple Random Walk bridge ( β = 0)? • Order of fluctuations. • e.g. µ σ n ,β ( � s ( n / 2 ) � 2 ) . • Pinning to certain regions in the environment. • Entropy vs. Energy. • Quenched vs. Annealed Results.
Introduction Results Idea of Proof Summary The Questions • For β > 0 does s ( · ) behaves macroscopically differently than a Simple Random Walk bridge ( β = 0)? • Order of fluctuations. • e.g. µ σ n ,β ( � s ( n / 2 ) � 2 ) . • Pinning to certain regions in the environment. • Entropy vs. Energy. • Quenched vs. Annealed Results.
Introduction Results Idea of Proof Summary Conjecture Theorem If d � 2 and β > 0 , or d > 2 and β ≫ 0 the polymer is super-diffusive with fluctuations of order n 2 / 3 . Otherwise, the polymer is diffusive - fluctuations of order n 1 / 2 . This is universal w.r.t a large class of distributions, including the i.i.d case with marginals that decay sufficiently fast. High disorder phase vs. low disorder phase.
Introduction Results Idea of Proof Summary Conjecture Theorem If d � 2 and β > 0 , or d > 2 and β ≫ 0 the polymer is super-diffusive with fluctuations of order n 2 / 3 . Otherwise, the polymer is diffusive - fluctuations of order n 1 / 2 . This is universal w.r.t a large class of distributions, including the i.i.d case with marginals that decay sufficiently fast. High disorder phase vs. low disorder phase.
Introduction Results Idea of Proof Summary Previous Results - Low Disorder Phase � n ,β ( � s ( n / 2 ) � 2 ) ∼ Cn 1 / 2 and even s ( n / 2 ) / • µ σ n / 2 ⇒ N ( 0 , I ) . • P -a.s. • d � 3, β ≪ ∞ , many distributions. • Imbrie-Spencer (88). Bolthausen (89). Song-Zhou (96). • lim n →∞ max y ∈ Z d µ σ n ,β ( s ( n / 2 ) = y ) = 0. • P -a.s. • d � 3, β ≪ ∞ , many distributions. • Carmona-Hu (02). Comets-Shiga-Yoshida (03).
Introduction Results Idea of Proof Summary Previous Results - Low Disorder Phase � n ,β ( � s ( n / 2 ) � 2 ) ∼ Cn 1 / 2 and even s ( n / 2 ) / • µ σ n / 2 ⇒ N ( 0 , I ) . • P -a.s. • d � 3, β ≪ ∞ , many distributions. • Imbrie-Spencer (88). Bolthausen (89). Song-Zhou (96). • lim n →∞ max y ∈ Z d µ σ n ,β ( s ( n / 2 ) = y ) = 0. • P -a.s. • d � 3, β ≪ ∞ , many distributions. • Carmona-Hu (02). Comets-Shiga-Yoshida (03).
Introduction Results Idea of Proof Summary Previous Results - High Disorder Phase • inf { ζ > 0 : µ σ n ,β ( � s � ∞ � n ζ ) → 1 in P -prob. } ∈ [ 3 / ( 4 + d ) , 3 / 4 ] . • d � 1. All β . Many distributions. • Not completely rigorous. • Piza (97) • lim sup n →∞ max y ∈ Z d µ σ n ,β ( s ( n / 2 ) = y ) > C . • P -a.s. • C is non-random. • d � 2 and β > 0 or d > 2 and β ≫ 0, many distributions. • Carmona-Hu (02). Comets-Shiga-Yoshida (03).
Introduction Results Idea of Proof Summary Previous Results - High Disorder Phase • inf { ζ > 0 : µ σ n ,β ( � s � ∞ � n ζ ) → 1 in P -prob. } ∈ [ 3 / ( 4 + d ) , 3 / 4 ] . • d � 1. All β . Many distributions. • Not completely rigorous. • Piza (97) • lim sup n →∞ max y ∈ Z d µ σ n ,β ( s ( n / 2 ) = y ) > C . • P -a.s. • C is non-random. • d � 2 and β > 0 or d > 2 and β ≫ 0, many distributions. • Carmona-Hu (02). Comets-Shiga-Yoshida (03).
Introduction Results Idea of Proof Summary Last/First Passage Percolation • If β = ∞ then µ σ n ,β is the δ measure on the maximal path � s n = arg max σ ( s ) . (uniform measure on maximal paths). • Model coincides with Last (first) Passage Percolation. • Supper-diffusive behavior and fluctuations of order n 2 / 3 for � s under the same universality class are conjectured as well. • Only partially proved. • Baik-Deift, Johansson, Licea-Newman-Piza, Newman-Pize.
Introduction Results Idea of Proof Summary Last/First Passage Percolation • If β = ∞ then µ σ n ,β is the δ measure on the maximal path � s n = arg max σ ( s ) . (uniform measure on maximal paths). • Model coincides with Last (first) Passage Percolation. • Supper-diffusive behavior and fluctuations of order n 2 / 3 for � s under the same universality class are conjectured as well. • Only partially proved. • Baik-Deift, Johansson, Licea-Newman-Piza, Newman-Pize.
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