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Directed polymer in -stable Random Environments Roberto Viveros IMPA. Rio de Janeiro-Brazil JULY 23, 2019 Roberto Viveros Directed polymer in -stable Random Environments Chemical background Roberto Viveros Directed polymer in -stable

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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

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Chemical background

Roberto Viveros Directed polymer in γ-stable Random Environments

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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

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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

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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

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Polymer Classification.

Homopolymers (polyethylene), Copolymers: periodic (agar), random (carrageenan).

Roberto Viveros Directed polymer in γ-stable Random Environments

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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

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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

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Polymerization

Size of a polymer the number of constituent monomers (also called the degree of polymerization) may vary from 103 up to 1010.

Roberto Viveros Directed polymer in γ-stable Random Environments

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Polymerization

Size of a polymer the number of constituent monomers (also called the degree of polymerization) may vary from 103 up to 1010. 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

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Polymerization

Size of a polymer the number of constituent monomers (also called the degree of polymerization) may vary from 103 up to 1010. 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

  • r can collapse to a ball due to attractive forces.

Roberto Viveros Directed polymer in γ-stable Random Environments

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Targets

Number of different spatial configurations

Roberto Viveros Directed polymer in γ-stable Random Environments

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Targets

Number of different spatial configurations Typical end-to-end distance (subdiffusive, diffusive or superdiffusive)

Roberto Viveros Directed polymer in γ-stable Random Environments

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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

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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

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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

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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

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Model Setting: Directed Polymer in Random Environment

Random Polymer Px: Probability measure on (Ω, F) :=

  • ZdN, P(Zd)⊗N
  • f

sequences S := (Sn)n≥0 such that: S0 = x, {Sn − Sn−1}n≥1 is an IID sequence, and Px[S1 = x + ej] = Px[S1 = x − ej] = 1 2d , (1)

Roberto Viveros Directed polymer in γ-stable Random Environments

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Model Setting: Directed Polymer in Random Environment

Random Environment IID random variables η := {ηn,z : n ∈ N, z ∈ Zd}, called the environment, defined on a probability space (Λ, F, P), that satisfies: E[η0,0] = 0 and E[exp(βη0,0)] < ∞, for all β ∈ R. (2)

Roberto Viveros Directed polymer in γ-stable Random Environments

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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 Pβ,η

N

  • n the space Ω, called the polymer

measure, by its Radon-Nikodym derivative with respect to P0: dPβ,η

N

dP0 (S) = 1 Z β,η

N

exp

  • β

N

  • n=1

ηn,Sn

  • ,

(3) where Z β,η

N

is the positive normalization factor that makes Pβ,η

N

a probability measure.

Roberto Viveros Directed polymer in γ-stable Random Environments

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Known results: Directed Polymer in Random Environment

Bolthausen ’87 W β,η

N

:= Z β,η

N

E

  • Z β,η

N

, (4) W β,η

∞

:= lim

N→∞ W β,η N ,

(5) exists P-a.s. and is a non-negative random variable.

Roberto Viveros Directed polymer in γ-stable Random Environments

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Known results: Directed Polymer in Random Environment

Bolthausen ’87 W β,η

N

:= Z β,η

N

E

  • Z β,η

N

, (4) W β,η

∞

:= lim

N→∞ W β,η N ,

(5) 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

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Weak disorder

Polymer paths have the same behavior as the simple random walk (delocalized phase)

Roberto Viveros Directed polymer in γ-stable Random Environments

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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 Pβ,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

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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

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Strong disorder

The polymer is largely influenced by the disorder and is attracted to sites with favorable environment (localize phase). Comtets, Shiga, Yoshida 2003

  • W β,η

∞

= 0

  • =

  

  • n≥1
  • Pβ,η

n−1

⊗2 [Sn = S′

n] = ∞

   P-a.s., (6) where S and S′ are two independent polymers with distribution Pβ,η

n−1. Moreover, if P[W β,η ∞

= 0] = 1, then there exists some constants c1, c2 ∈ (0, ∞) such that, −c1 log W β,η

N

≤

N

  • n≥1
  • Pβ,η

n−1

⊗2 [Sn = S′

n] ≤ −c2 log W β,η N

(7)

Roberto Viveros Directed polymer in γ-stable Random Environments

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Phase Transition

Comets, Yoshida 2005 There exists a critical value βc = βc(d) ∈ [0, ∞] with βc = 0 for d = 1, 2 and (8) βc > 0 for d ≥ 3, (9) such that there is weak disorder for β ∈ [0, βc) and strong disorder for β > βc.

Roberto Viveros Directed polymer in γ-stable Random Environments

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L2 region

Example: Showing weak disorder for d ≥ 3 and small β. E

  • W β,η

N

2 = EE⊗2  exp  

i≥N

βηi,Si − β2

2 + βηi,S′

i − β2

2

    (10) ≤ E⊗2

  • exp
  • i

β21{Si=S′

i }

  • .

(11)

Roberto Viveros Directed polymer in γ-stable Random Environments

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L2 region

Example: Showing weak disorder for d ≥ 3 and small β. E

  • W β,η

N

2 = EE⊗2  exp  

i≥N

βηi,Si − β2

2 + βηi,S′

i − β2

2

    (10) ≤ E⊗2

  • exp
  • i

β21{Si=S′

i }

  • .

(11) Then,

  • W β,η

N

  • N is uniformly integrable.

EW β,η

∞

= limN→∞ EW β,η

N

= 1

Roberto Viveros Directed polymer in γ-stable Random Environments

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Changing the setup

Roberto Viveros Directed polymer in γ-stable Random Environments

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Changing the setup

Our polymer measure dPβ,ω

N

dP0 (S) = 1 Z β,ω

N

N

  • n=1

(1 + βωn,Sn)

  • ,

(12)

Roberto Viveros Directed polymer in γ-stable Random Environments

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Changing the setup

Our polymer measure dPβ,ω

N

dP0 (S) = 1 Z β,ω

N

N

  • n=1

(1 + βωn,Sn)

  • ,

(12) Assuming, ω0,0 ≥ −1 P-a.s., E[ω0,0] = 0 and P[ω0,0 > x] x→∞ ∼ CPx−γ, for γ ∈ (1, 2), (13)

Roberto Viveros Directed polymer in γ-stable Random Environments

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Our results

Free energy, p(β) := limN→∞ 1

N Z β,ω N

Roberto Viveros Directed polymer in γ-stable Random Environments

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Our results

Free energy, p(β) := limN→∞ 1

N Z β,ω N

There is a critical value γc = γc(d) := 1 + 2 d .

Roberto Viveros Directed polymer in γ-stable Random Environments

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Our results

Free energy, p(β) := limN→∞ 1

N Z β,ω N

There is a critical value γc = γc(d) := 1 + 2 d . Theorem 1 When the environment satisfies the condition above and if γ ≤ γc, very strong disorder holds, for all values of β > 0, in all dimensions d ≥ 1. In particular, if d ≥ 3 and γ < γc, lim

β→0

log |p(β)| log β = α, (14) where α = α(d, γ) := γ(γc−1)

γc−γ . Also, if γ = γc, we have that,

lim

β→0

log |p(β)| log β = ∞. (15)

Roberto Viveros Directed polymer in γ-stable Random Environments

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Our results

Roberto Viveros Directed polymer in γ-stable Random Environments

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Our results

Theorem 2 Assuming the same conditions for the environment and if γ > γc, we have that βc > 0 for dimensions d ≥ 3.

Roberto Viveros Directed polymer in γ-stable Random Environments