Directed polymer in γ -stable Random Environments Roberto Viveros IMPA. Rio de Janeiro-Brazil JULY 23, 2019 Roberto Viveros Directed polymer in γ -stable Random Environments
Chemical background Roberto Viveros Directed polymer in γ -stable Random Environments
Chemical background A polymer is a large molecule consisting of monomers that are tied together by chemical bonds. Roberto Viveros Directed polymer in γ -stable Random Environments
Chemical background A polymer is a large molecule consisting of monomers that are tied together by chemical bonds. The monomers can be either small units, such as CH2 in polyethylene; or larger units with an internal structure (such as the adenine-thymine and cytosine-guanine base pairs in the DNA double helix. Roberto Viveros Directed polymer in γ -stable Random Environments
Chemical background A polymer is a large molecule consisting of monomers that are tied together by chemical bonds. The monomers can be either small units, such as CH2 in polyethylene; or larger units with an internal structure (such as the adenine-thymine and cytosine-guanine base pairs in the DNA double helix. Polymers abound in nature because of the multivalency of atoms like carbon, silicon, oxygen, nitrogen, sulfur and phosphorus, which are capable of forming long concatenated structures. Roberto Viveros Directed polymer in γ -stable Random Environments
Polymer Classification. Homopolymers (polyethylene), Copolymers: periodic (agar), random (carrageenan). Roberto Viveros Directed polymer in γ -stable Random Environments
Polymer Classification. Homopolymers (polyethylene), Copolymers: periodic (agar), random (carrageenan). Synthetic (nylon), Natural: proteins (amino-acids), nucleic acids (DNA, RNA), polysaccharides (amylose, cellulose), lignin (plant cement), rubber (fluid of latex cells). Roberto Viveros Directed polymer in γ -stable Random Environments
Polymer Classification. Homopolymers (polyethylene), Copolymers: periodic (agar), random (carrageenan). Synthetic (nylon), Natural: proteins (amino-acids), nucleic acids (DNA, RNA), polysaccharides (amylose, cellulose), lignin (plant cement), rubber (fluid of latex cells). linear, branched. Roberto Viveros Directed polymer in γ -stable Random Environments
Polymerization Size of a polymer the number of constituent monomers (also called the degree of polymerization) may vary from 10 3 up to 10 10 . Roberto Viveros Directed polymer in γ -stable Random Environments
Polymerization Size of a polymer the number of constituent monomers (also called the degree of polymerization) may vary from 10 3 up to 10 10 . the polymer can arrange itself in many different spatial configurations can wind around itself to form knots can be extended due to repulsive forces between the monomers as a result of excluded-volume Roberto Viveros Directed polymer in γ -stable Random Environments
Polymerization Size of a polymer the number of constituent monomers (also called the degree of polymerization) may vary from 10 3 up to 10 10 . the polymer can arrange itself in many different spatial configurations can wind around itself to form knots can be extended due to repulsive forces between the monomers as a result of excluded-volume or can collapse to a ball due to attractive forces. Roberto Viveros Directed polymer in γ -stable Random Environments
Targets Number of different spatial configurations Roberto Viveros Directed polymer in γ -stable Random Environments
Targets Number of different spatial configurations Typical end-to-end distance (subdiffusive, diffusive or superdiffusive) Roberto Viveros Directed polymer in γ -stable Random Environments
Targets Number of different spatial configurations Typical end-to-end distance (subdiffusive, diffusive or superdiffusive) Space-time scaling limit Roberto Viveros Directed polymer in γ -stable Random Environments
Targets Number of different spatial configurations Typical end-to-end distance (subdiffusive, diffusive or superdiffusive) Space-time scaling limit Average length and height of excursions away from an interface or a surface. Roberto Viveros Directed polymer in γ -stable Random Environments
Targets Number of different spatial configurations Typical end-to-end distance (subdiffusive, diffusive or superdiffusive) Space-time scaling limit Average length and height of excursions away from an interface or a surface. Free energy of the polymer in this limit, Roberto Viveros Directed polymer in γ -stable Random Environments
Targets Number of different spatial configurations Typical end-to-end distance (subdiffusive, diffusive or superdiffusive) Space-time scaling limit Average length and height of excursions away from an interface or a surface. Free energy of the polymer in this limit, presence of phase transitions as a function of underlying model parameters Roberto Viveros Directed polymer in γ -stable Random Environments
Model Setting: Directed Polymer in Random Environment Random Polymer �� Z d � N , P ( Z d ) ⊗ N � P x : Probability measure on (Ω , F ) := of sequences S := ( S n ) n ≥ 0 such that: S 0 = x , { S n − S n − 1 } n ≥ 1 is an IID sequence, and (1) P x [ S 1 = x + e j ] = P x [ S 1 = x − e j ] = 1 2 d , Roberto Viveros Directed polymer in γ -stable Random Environments
Model Setting: Directed Polymer in Random Environment Random Environment IID random variables η := { η n , z : n ∈ N , z ∈ Z d } , called the environment , defined on a probability space (Λ , F , P ), that satisfies: E [ η 0 , 0 ] = 0 and (2) E [exp( βη 0 , 0 )] < ∞ , for all β ∈ R . Roberto Viveros Directed polymer in γ -stable Random Environments
Model Setting: Directed Polymer in Random Environment Polymer Measure Given β > 0, N ∈ N and a fixed realization of the environment η , we define the measure P β,η on the space Ω, called the polymer N measure , by its Radon-Nikodym derivative with respect to P 0 : � N � d P β,η 1 � N ( S ) = exp (3) β η n , S n , Z β,η d P 0 n =1 N where Z β,η is the positive normalization factor that makes P β,η a N N probability measure. Roberto Viveros Directed polymer in γ -stable Random Environments
Known results: Directed Polymer in Random Environment Bolthausen ’87 Z β,η W β,η N := � , (4) N � Z β,η E N N →∞ W β,η W β,η := lim (5) N , ∞ exists P - a.s. and is a non-negative random variable. Roberto Viveros Directed polymer in γ -stable Random Environments
Known results: Directed Polymer in Random Environment Bolthausen ’87 Z β,η W β,η N := � , (4) N � Z β,η E N N →∞ W β,η W β,η := lim (5) N , ∞ exists P - a.s. and is a non-negative random variable. we have weak disorder if W β ∞ > 0 P - a.s. and strong disorder if W β ∞ = 0 P - a.s. . Roberto Viveros Directed polymer in γ -stable Random Environments
Weak disorder Polymer paths have the same behavior as the simple random walk (delocalized phase) Roberto Viveros Directed polymer in γ -stable Random Environments
Weak disorder Polymer paths have the same behavior as the simple random walk (delocalized phase) Comets, Yoshida 2005 Assuming d ≥ 3 and weak disorder, the measures P β,η N , after rescaling, converge in law to the Brownian Motion, for almost all realizations of the environment. Roberto Viveros Directed polymer in γ -stable Random Environments
Strong disorder The polymer is largely influenced by the disorder and is attracted to sites with favorable environment (localize phase). Roberto Viveros Directed polymer in γ -stable Random Environments
Strong disorder The polymer is largely influenced by the disorder and is attracted to sites with favorable environment (localize phase). Comtets, Shiga, Yoshida 2003 � ⊗ 2 � � � W β,η � P β,η [ S n = S ′ = 0 = n ] = ∞ P -a.s., (6) ∞ n − 1 n ≥ 1 where S and S ′ are two independent polymers with distribution P β,η n − 1 . Moreover, if P [ W β,η = 0] = 1, then there exists some ∞ constants c 1 , c 2 ∈ (0 , ∞ ) such that, N � ⊗ 2 � − c 1 log W β,η � P β,η n ] ≤ − c 2 log W β,η [ S n = S ′ ≤ (7) n − 1 N N n ≥ 1 Roberto Viveros Directed polymer in γ -stable Random Environments
Phase Transition Comets, Yoshida 2005 There exists a critical value β c = β c ( d ) ∈ [0 , ∞ ] with β c = 0 for d = 1 , 2 and (8) β c > 0 for d ≥ 3 , (9) such that there is weak disorder for β ∈ [0 , β c ) and strong disorder for β > β c . Roberto Viveros Directed polymer in γ -stable Random Environments
L 2 region Example: Showing weak disorder for d ≥ 3 and small β . �� � 2 � βη i , S i − β 2 i − β 2 W β,η � = E E ⊗ 2 exp 2 + βη i , S ′ E N 2 i ≥ N (10) � �� �� ≤ E ⊗ 2 β 2 1 { S i = S ′ exp (11) . i } i Roberto Viveros Directed polymer in γ -stable Random Environments
Recommend
More recommend
