Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
A directed polymer approach to the once-oriented
last passage site percolation time constant in
high dimensions
Gregory J. Morrow
Department of Mathematics
University of Colorado at Colorado Springs
Conference in Memory of Walter V. Philipp
June, 2009
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Directed path for d = 1.
There are (2d)n directed paths of length n, with height of path y ∈ Zd .
·
y ∈ Z1
·
·
@
@·
@
@
·
·
@
@· @
@·
·
%@
@
&
@
@
·
··
%· &
@
&
@· @
@·
@· @
@
@
@·
@·
@
@· @
@·
@
@·
@
@·
t
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
“Point to Line” Last Passage Times
I
Nearest Neighbor “heights” in Zd :
γt , t = 0, 1, 2, . . . , n (γ(0) = 0).
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
“Point to Line” Last Passage Times
I
Nearest Neighbor “heights” in Zd :
I
γt , t = 0, 1, 2, . . . , n (γ(0) = 0).
−
Directed Path: →
γn := (t, γt ), t = 0, 1, 2, . . . , n.
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
“Point to Line” Last Passage Times
I
I
I
Nearest Neighbor “heights” in Zd :
γt , t = 0, 1, 2, . . . , n (γ(0) = 0).
−
Directed Path: →
γn := (t, γt ), t = 0, 1, 2, . . . , n.
I.I.D. Weights, η, at sites (t, y ), with
E(|η|) < ∞ and (WLOG) Eη ≥ 0. Also called “passage
times”, η(t, y ).
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
“Point to Line” Last Passage Times
I
I
I
I
Nearest Neighbor “heights” in Zd :
γt , t = 0, 1, 2, . . . , n (γ(0) = 0).
−
Directed Path: →
γn := (t, γt ), t = 0, 1, 2, . . . , n.
I.I.D. Weights, η, at sites (t, y ), with
E(|η|) < ∞ and (WLOG) Eη ≥ 0. Also called “passage
times”, η(t, y ).
“Passage Time” along a directed path:
n
X
→
−
T (γn ) :=
η(t, γt ).
t=1
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
“Point to Line” Last Passage Times
I
I
I
I
Nearest Neighbor “heights” in Zd :
γt , t = 0, 1, 2, . . . , n (γ(0) = 0).
−
Directed Path: →
γn := (t, γt ), t = 0, 1, 2, . . . , n.
I.I.D. Weights, η, at sites (t, y ), with
E(|η|) < ∞ and (WLOG) Eη ≥ 0. Also called “passage
times”, η(t, y ).
“Passage Time” along a directed path:
n
X
→
−
T (γn ) :=
η(t, γt ).
t=1
I
“Point to Point” Last Passage Time:
−
e0,n := max T (→
T
γn ).
γ0 =γn =0
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
“Point to Line” Last Passage Time Constant
I
Superadditivity of “Point to Point” Last Passage Times:
e0,n+m ≥ T
e0,m + T
em,m+n , so, by Kingman,
T
e0,n /n = µd , P-a.s. and in L1 .
lim T
n→∞
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
“Point to Line” Last Passage Time Constant
I
Superadditivity of “Point to Point” Last Passage Times:
e0,n+m ≥ T
e0,m + T
em,m+n , so, by Kingman,
T
e0,n /n = µd , P-a.s. and in L1 .
lim T
n→∞
I
“Point to Line” Last Passage Times:P
e 0,n := maxγ =0 n η(t, γt ).
H
0
t=1
e 0,n } is a superadditive sequence : EH
e 0,n+m ≥ EH
e 0,n +EH
e 0,m .
{EH
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
“Point to Line” Last Passage Time Constant
I
Superadditivity of “Point to Point” Last Passage Times:
e0,n+m ≥ T
e0,m + T
em,m+n , so, by Kingman,
T
e0,n /n = µd , P-a.s. and in L1 .
lim T
n→∞
I
“Point to Line” Last Passage Times:P
e 0,n := maxγ =0 n η(t, γt ).
H
0
t=1
e 0,n } is a superadditive sequence : EH
e 0,n+m ≥ EH
e 0,n +EH
e 0,m .
{EH
I
“Point to Line” Last Passage Time Constant:
e 0,n /n = sup EH
e 0,n /n.
νd = νd (η) := lim EH
n→∞
n
.
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
“Point to Line” Last Passage Time Constant
I
Superadditivity of “Point to Point” Last Passage Times:
e0,n+m ≥ T
e0,m + T
em,m+n , so, by Kingman,
T
e0,n /n = µd , P-a.s. and in L1 .
lim T
n→∞
I
“Point to Line” Last Passage Times:P
e 0,n := maxγ =0 n η(t, γt ).
H
0
t=1
e 0,n } is a superadditive sequence : EH
e 0,n+m ≥ EH
e 0,n +EH
e 0,m .
{EH
I
“Point to Line” Last Passage Time Constant:
e 0,n /n = sup EH
e 0,n /n.
νd = νd (η) := lim EH
n→∞
I
n
.
By Hammersley and Welsh: νd = µd .
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Evaluate ν1 for Exponential and Geometric cases
I
If η is unit exponential then:
ν1 (η) = 2.
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Evaluate ν1 for Exponential and Geometric cases
I
If η is unit exponential then:
ν1 (η) = 2.
I
If η is geometric (p), then:
p
ν1 (η) = (1 + 1 − p)/p.
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Evaluate ν1 for Exponential and Geometric cases
I
If η is unit exponential then:
ν1 (η) = 2.
I
If η is geometric (p), then:
p
ν1 (η) = (1 + 1 − p)/p.
I
For both these special cases a Shape Theorem holds:
e0,(nx,ny ) = g (x, y )
lim (1/n)T
n→∞
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Evaluate ν1 for Exponential and Geometric cases
I
If η is unit exponential then:
ν1 (η) = 2.
I
If η is geometric (p), then:
p
ν1 (η) = (1 + 1 − p)/p.
I
For both these special cases a Shape Theorem holds:
e0,(nx,ny ) = g (x, y )
lim (1/n)T
n→∞
I
e0,n − nν1 )/n1/3 converges in
Also in these special cases (T
distribution to the Tracy-Widom law F2 [ref. K. Johansson
CMP (2000), relations to random matrix theory]
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Directed Polymer: Partition Function and Free Energy
I
Directed paths are polymers. Higher total weights are favored.
Partition function:
X
−
Zn (β) := (2d)−n
exp(βT (→
γn )).
γ
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Directed Polymer: Partition Function and Free Energy
I
Directed paths are polymers. Higher total weights are favored.
Partition function:
X
−
Zn (β) := (2d)−n
exp(βT (→
γn )).
γ
I
−
−
Gibbs measure: µn (→
γ ) := (2d)−n exp(βT (→
γn ))/Zn (β).
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Directed Polymer: Partition Function and Free Energy
I
Directed paths are polymers. Higher total weights are favored.
Partition function:
X
−
Zn (β) := (2d)−n
exp(βT (→
γn )).
γ
I
I
−
−
Gibbs measure: µn (→
γ ) := (2d)−n exp(βT (→
γn ))/Zn (β).
By Jensen’s inequality, {E ln Zn } is superadditive; so define
the Free Energy:
f (β) := lim (1/n)E ln Zn = sup(1/n)E ln Zn
n→∞
n
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Directed Polymer: Partition Function and Free Energy
I
Directed paths are polymers. Higher total weights are favored.
Partition function:
X
−
Zn (β) := (2d)−n
exp(βT (→
γn )).
γ
I
I
−
−
Gibbs measure: µn (→
γ ) := (2d)−n exp(βT (→
γn ))/Zn (β).
By Jensen’s inequality, {E ln Zn } is superadditive; so define
the Free Energy:
f (β) := lim (1/n)E ln Zn = sup(1/n)E ln Zn
n→∞
I
n
Log Moment Generating function: λ(β) := ln E exp(βη)
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Directed Polymer: Partition Function and Free Energy
I
Directed paths are polymers. Higher total weights are favored.
Partition function:
X
−
Zn (β) := (2d)−n
exp(βT (→
γn )).
γ
I
I
−
−
Gibbs measure: µn (→
γ ) := (2d)−n exp(βT (→
γn ))/Zn (β).
By Jensen’s inequality, {E ln Zn } is superadditive; so define
the Free Energy:
f (β) := lim (1/n)E ln Zn = sup(1/n)E ln Zn
n→∞
I
I
n
Log Moment Generating function: λ(β) := ln E exp(βη)
By Jensen’s inequality w.r.t. E again, f (β) ≤ λ(β).
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Properties of f (β)
I
fn (β) := E(ln Zn )/n is convex, because fn00 (β) is a variance
relative to Gibbs measure. Therefore f (β) is convex.
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Properties of f (β)
I
fn (β) := E(ln Zn )/n is convex, because fn00 (β) is a variance
relative to Gibbs measure. Therefore f (β) is convex.
I
Further, since E(η) ≥ 0, fn (β) is non-decreasing. Indeed,
for β1 > β0 ,
fn (β1 ) = fn (β0 )+
(1/n)E ln E exp(∆βT ) [exp(β0 T )/E exp(β0 T )]
e exp(∆βT ) ≥ fn (β0 )+EE(∆β)T
e
= fn (β0 )+(1/n)E ln E
/n ≥ fn (β0 ),
where at the last step Jensen’s inequality was applied w.r.t.
e Here E is the uniform measure on directed paths and E
e is
E.
simply the Gibbs measure expectation.
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Proposition 1
I
Estimate
P
γ
−
exp(βT (→
γn )) from below by the maximum term:
E(ln Zn ) = E ln[(2d)−n
X
−
e 0,n − n ln(2d)
exp(βT (→
γn ))] ≥ βEH
γ
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Proposition 1
I
Estimate
P
γ
−
exp(βT (→
γn )) from below by the maximum term:
E(ln Zn ) = E ln[(2d)−n
X
−
e 0,n − n ln(2d)
exp(βT (→
γn ))] ≥ βEH
γ
I
Estimate also the average above by the largest term:
−
e 0,n = max(βT (→
βEH
γn )) ≥ E(ln Zn ).
γ
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Proposition 1
I
Estimate
P
γ
−
exp(βT (→
γn )) from below by the maximum term:
E(ln Zn ) = E ln[(2d)−n
X
−
e 0,n − n ln(2d)
exp(βT (→
γn ))] ≥ βEH
γ
I
Estimate also the average above by the largest term:
−
e 0,n = max(βT (→
βEH
γn )) ≥ E(ln Zn ).
γ
I
By dividing by n and taking n → ∞, we have thus proved:
Proposition 1:
f (β) ≥ βνd − ln(2d), and lim f (β)/β = νd .
β→∞
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Prototypical examples
I
Exponential case: λ(β) = − ln(1 − β) ≥ 0, 0 ≤ β < 1.
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Prototypical examples
I
Exponential case: λ(β) = − ln(1 − β) ≥ 0, 0 ≤ β < 1.
I
Poisson case: λ(β) = exp(β) − 1 ≥ 0, 0 ≤ β < ∞.
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Prototypical examples
I
Exponential case: λ(β) = − ln(1 − β) ≥ 0, 0 ≤ β < 1.
I
Poisson case: λ(β) = exp(β) − 1 ≥ 0, 0 ≤ β < ∞.
I
Gaussian case: λ(β) = β 2 /2, 0 ≤ β < ∞.
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Prototypical examples
I
Exponential case: λ(β) = − ln(1 − β) ≥ 0, 0 ≤ β < 1.
I
Poisson case: λ(β) = exp(β) − 1 ≥ 0, 0 ≤ β < ∞.
I
Gaussian case: λ(β) = β 2 /2, 0 ≤ β < ∞.
I
Generic case: P(η > x) = exp(−x · x α ), x ≥ 0, some α > 0.
λ(β) ∼ Cα β 1+1/α .
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Prototypical examples
I
Exponential case: λ(β) = − ln(1 − β) ≥ 0, 0 ≤ β < 1.
I
Poisson case: λ(β) = exp(β) − 1 ≥ 0, 0 ≤ β < ∞.
I
Gaussian case: λ(β) = β 2 /2, 0 ≤ β < ∞.
I
Generic case: P(η > x) = exp(−x · x α ), x ≥ 0, some α > 0.
λ(β) ∼ Cα β 1+1/α .
I
When λ(β) exists for all β we know by the martingale theory
first developed by Bolthausen CMP(1989), and later by
Comets-Yoshida AP (2006) that for high d, λ(β) = f (β) for
small β > 0. We essentially determine in a generic case for
high d where the departure of the two curves λ(β) and f (β)
occurs in terms of d.
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
The Gaussian case
I
Carmona and Hu PTRF (2002) have shown in the Gaussian
case (λ(β) = β 2 /2) that :
p
p
f (β) ≤ min β 2 /2, β 2 ln(2d) , so, νd ≤ 2 ln(2d)
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
The Gaussian case
I
Carmona and Hu PTRF (2002) have shown in the Gaussian
case (λ(β) = β 2 /2) that :
p
p
f (β) ≤ min β 2 /2, β 2 ln(2d) , so, νd ≤ 2 ln(2d)
I
Obviously, putting md := E max(η1 , . . . , η2d ),
e 0,n ≥ EH
e 0,1 = md .
νd = sup(1/n)EH
n≥1
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
The Gaussian case
I
Carmona and Hu PTRF (2002) have shown in the Gaussian
case (λ(β) = β 2 /2) that :
p
p
f (β) ≤ min β 2 /2, β 2 ln(2d) , so, νd ≤ 2 ln(2d)
I
Obviously, putting md := E max(η1 , . . . , η2d ),
e 0,n ≥ EH
e 0,1 = md .
νd = sup(1/n)EH
n≥1
I
But one can show directly in the Gaussian case that
p
md = 2 ln(2d) + o(1)
p
Hence in this case: νd = 2 ln(2d) + o(1), as d → ∞.
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Exponential case: νd & ln(2d), as d → ∞
I
We have νd ≥ md . We first estimate
md := E max(η1 , . . . , η2d ) from below.
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Exponential case: νd & ln(2d), as d → ∞
I
I
We have νd ≥ md . We first estimate
md := E max(η1 , . . . , η2d ) from below.
Write G (t) := t 2d for t = t(x) := P(η ≤ x) = 1 − exp(−x).
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Exponential case: νd & ln(2d), as d → ∞
I
I
I
We have νd ≥ md . We first estimate
md := E max(η1 , . . . , η2d ) from below.
Write G (t) := t 2d for t = t(x) := P(η ≤ x) = 1 − exp(−x).
By the change of variables t = t(x) we have
Z ∞
Z 1
2d
md =
xdt (x) =
− ln(1 − t)dG (t).
0
0
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Exponential case: νd & ln(2d), as d → ∞
I
I
I
We have νd ≥ md . We first estimate
md := E max(η1 , . . . , η2d ) from below.
Write G (t) := t 2d for t = t(x) := P(η ≤ x) = 1 − exp(−x).
By the change of variables t = t(x) we have
Z ∞
Z 1
2d
md =
xdt (x) =
− ln(1 − t)dG (t).
0
I
0
But, − ln is a convex function. So by Jensen’s inequality,
Z 1
md ≥ − ln( (1 − t)dG (t)) = − ln(1 − 2d/(2d + 1))
0
= ln(2d) + o(1), as d → ∞.
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Exponential case: νd . ln(2d), as d → ∞
I
Next we apply Proposition 1:
λ(β) ≥ f (β) ≥ βνd − ln(2d).
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Exponential case: νd . ln(2d), as d → ∞
I
Next we apply Proposition 1:
λ(β) ≥ f (β) ≥ βνd − ln(2d).
I
Now λ(β) = − ln(1 − β), β < 1. Choose β = 1 − 1/ ln(2d).
We get that
ln ln(2d) ≥ νd (1 − 1/ ln(2d)) − ln(2d).
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Exponential case: νd . ln(2d), as d → ∞
I
Next we apply Proposition 1:
λ(β) ≥ f (β) ≥ βνd − ln(2d).
I
Now λ(β) = − ln(1 − β), β < 1. Choose β = 1 − 1/ ln(2d).
We get that
ln ln(2d) ≥ νd (1 − 1/ ln(2d)) − ln(2d).
I
It follows that
νd ≤ ln(2d) + O(ln ln(2d))
= ln(2d)(1 + o(1)), as d → ∞.
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Exponential case: νd . ln(2d), as d → ∞
I
Next we apply Proposition 1:
λ(β) ≥ f (β) ≥ βνd − ln(2d).
I
Now λ(β) = − ln(1 − β), β < 1. Choose β = 1 − 1/ ln(2d).
We get that
ln ln(2d) ≥ νd (1 − 1/ ln(2d)) − ln(2d).
I
It follows that
νd ≤ ln(2d) + O(ln ln(2d))
= ln(2d)(1 + o(1)), as d → ∞.
I
By the previous page we therefore have
νd ∼ ln(2d), as d → ∞.
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Assumptions to Construct a Lower Bound for md
I
Write F (x) := P(η ≤ x). Assume
F (x) = 1−exp(−u(x)), for u(x) non-decreasing to ∞, as x → ∞.
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Assumptions to Construct a Lower Bound for md
I
Write F (x) := P(η ≤ x). Assume
F (x) = 1−exp(−u(x)), for u(x) non-decreasing to ∞, as x → ∞.
I
Also, because u(x) need not be continuous, assume there are
exponents
u0 (x) ≤ u(x) ≤ u1 (x), with ui0 (x) > 0, and u1 (x) = (1+o(1))u0 (x)
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Assumptions to Construct a Lower Bound for md
I
Write F (x) := P(η ≤ x). Assume
F (x) = 1−exp(−u(x)), for u(x) non-decreasing to ∞, as x → ∞.
I
Also, because u(x) need not be continuous, assume there are
exponents
u0 (x) ≤ u(x) ≤ u1 (x), with ui0 (x) > 0, and u1 (x) = (1+o(1))u0 (x)
I
Assume that Fi (x) := 1 − exp(−ui (x)) is concave for large x,
each i = 0, 1.
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Assumptions to Construct a Lower Bound for md
I
Write F (x) := P(η ≤ x). Assume
F (x) = 1−exp(−u(x)), for u(x) non-decreasing to ∞, as x → ∞.
I
Also, because u(x) need not be continuous, assume there are
exponents
u0 (x) ≤ u(x) ≤ u1 (x), with ui0 (x) > 0, and u1 (x) = (1+o(1))u0 (x)
I
I
Assume that Fi (x) := 1 − exp(−ui (x)) is concave for large x,
each i = 0, 1.
Finally, assume the exponential tail condition
lim inf ui0 (x) > 0, i = 0, 1.
x→∞
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Lower bound for md
I
Define U(u) as the inverse function of u: U := u −1 .
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Lower bound for md
I
I
Define U(u) as the inverse function of u: U := u −1 .
Lemma 1:
md & U(ln(2d)), as d → ∞.
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Lower bound for md
I
I
I
Define U(u) as the inverse function of u: U := u −1 .
Lemma 1:
md & U(ln(2d)), as d → ∞.
Proof uses that Fi−1 (t) is convex, i = 0, 1 and “brackets”
F −1 (t). In particular, by Jensen’s inequality
Z 1
Z 1
−1
2d
md =
F (t)d(t ) &
F1−1 (t)d t 2d /(1 − t02d )
0
t0
& F1−1 (1 − 1/(2d) + O(1/d 2 )).
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Lower bound for md
I
I
I
Define U(u) as the inverse function of u: U := u −1 .
Lemma 1:
md & U(ln(2d)), as d → ∞.
Proof uses that Fi−1 (t) is convex, i = 0, 1 and “brackets”
F −1 (t). In particular, by Jensen’s inequality
Z 1
Z 1
−1
2d
md =
F (t)d(t ) &
F1−1 (t)d t 2d /(1 − t02d )
0
t0
& F1−1 (1 − 1/(2d) + O(1/d 2 )).
I
We obtain, using that U1 := u1−1 grows at most linearly,
F1−1 (1−1/(2d)+O(1/d 2 )) = U1 (ln(2d)+o(1)) ∼ U1 (ln(2d)).
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
A Class of Distributions
I
Now assume u(x) = xv (x) + δ(x), x ≥ x0 [cf. Ben-Ari
(2007)], where δ(x) = O(x), and v (x) is twice continuously
differentiable and satisfies the following shape, growth, and
regularity conditions:
v (x), x ≥ x0 % ∞, and xv (x), x ≥ x0 , is strictly convex
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
A Class of Distributions
I
Now assume u(x) = xv (x) + δ(x), x ≥ x0 [cf. Ben-Ari
(2007)], where δ(x) = O(x), and v (x) is twice continuously
differentiable and satisfies the following shape, growth, and
regularity conditions:
v (x), x ≥ x0 % ∞, and xv (x), x ≥ x0 , is strictly convex
I
growth:
lim inf x→∞ xv 0 (x) > 0.
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
A Class of Distributions
I
Now assume u(x) = xv (x) + δ(x), x ≥ x0 [cf. Ben-Ari
(2007)], where δ(x) = O(x), and v (x) is twice continuously
differentiable and satisfies the following shape, growth, and
regularity conditions:
v (x), x ≥ x0 % ∞, and xv (x), x ≥ x0 , is strictly convex
I
growth:
I
regularity:
lim inf x→∞ xv 0 (x) > 0.
lim inf x→∞ (xv (x))02 /(xv (x))00 > 1.
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
A Class of Distributions
I
Now assume u(x) = xv (x) + δ(x), x ≥ x0 [cf. Ben-Ari
(2007)], where δ(x) = O(x), and v (x) is twice continuously
differentiable and satisfies the following shape, growth, and
regularity conditions:
v (x), x ≥ x0 % ∞, and xv (x), x ≥ x0 , is strictly convex
lim inf x→∞ xv 0 (x) > 0.
I
growth:
I
regularity:
I
Theorem 1. Under these conditions,
lim inf x→∞ (xv (x))02 /(xv (x))00 > 1.
νd ∼ U(ln(2d)), as d → ∞.
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Proof of Thm 1 by Log MGF and Proposition 1
I
Define a random variable η+ taking values in [x0 , ∞) to have
the distribution function F+ (x) := P(η ≤ x|η ≥ x0 ), so
λ+ (β) ≥ f+ (β) ≥ νd (η+ )β − ln(2d) ≥ νd (η)β − ln(2d)
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Proof of Thm 1 by Log MGF and Proposition 1
I
Define a random variable η+ taking values in [x0 , ∞) to have
the distribution function F+ (x) := P(η ≤ x|η ≥ x0 ), so
λ+ (β) ≥ f+ (β) ≥ νd (η+ )β − ln(2d) ≥ νd (η)β − ln(2d)
I
Lemma 2. Define w (x) := (xv (x))0 = xv 0 (x) + v (x). Put
U = Ud = U(ln(2d)) for U = the inverse of xv (x), and
β = βd := w (U) − M for a constant M > 0.Then
λ+ (βd ) . U 2 v 0 (U), as d → ∞.
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Proof of Thm 1 by Log MGF and Proposition 1
I
Define a random variable η+ taking values in [x0 , ∞) to have
the distribution function F+ (x) := P(η ≤ x|η ≥ x0 ), so
λ+ (β) ≥ f+ (β) ≥ νd (η+ )β − ln(2d) ≥ νd (η)β − ln(2d)
I
Lemma 2. Define w (x) := (xv (x))0 = xv 0 (x) + v (x). Put
U = Ud = U(ln(2d)) for U = the inverse of xv (x), and
β = βd := w (U) − M for a constant M > 0.Then
λ+ (βd ) . U 2 v 0 (U), as d → ∞.
I
Finally, by these two estimates,
ln(2d) + U 2 v 0 (U) & w (U)νd , or, u(U) + U 2 v 0 (U) & w (U)νd
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Proof of Thm 1 by Log MGF and Proposition 1
I
Define a random variable η+ taking values in [x0 , ∞) to have
the distribution function F+ (x) := P(η ≤ x|η ≥ x0 ), so
λ+ (β) ≥ f+ (β) ≥ νd (η+ )β − ln(2d) ≥ νd (η)β − ln(2d)
I
Lemma 2. Define w (x) := (xv (x))0 = xv 0 (x) + v (x). Put
U = Ud = U(ln(2d)) for U = the inverse of xv (x), and
β = βd := w (U) − M for a constant M > 0.Then
λ+ (βd ) . U 2 v 0 (U), as d → ∞.
I
Finally, by these two estimates,
ln(2d) + U 2 v 0 (U) & w (U)νd , or, u(U) + U 2 v 0 (U) & w (U)νd
I
But u(x) + x 2 v 0 (x) = xw (x), so U & νd .
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Comments on the Proof
I
Since by the lower bound of Lemma 1, νd & U(ln(2d)), and
as a consequence of Proposition 1 and Lemma 2, we have the
upper bound νd . U(ln(2d)), the Theorem is proved.
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Comments on the Proof
I
I
Since by the lower bound of Lemma 1, νd & U(ln(2d)), and
as a consequence of Proposition 1 and Lemma 2, we have the
upper bound νd . U(ln(2d)), the Theorem is proved.
Comments on the proof of Lemma 2. After integration by
parts, we have the formula:
Z ∞
exp(λ+ (β)) = C0 β
exp(βx − xv (x) + δ(x))dx.
x0
R V (2β) R ∞
Split the integral by x0
+ V (2β) , where V := v −1 . The
second integral is negligible since for x ≥ V (2β), we have
x(β − v (x)) + δ(x) ≤ (M − β)x ≤ −(β/2)x
.
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Comments on the Proof
I
The main contribution of exp(λ+ (β)) is thus bounded above
by
C0 β exp
max {x(β + M) − xv (x)} [V (2β) − x0 ]
x0 ≤x≤V (2β)
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
Comments on the Proof
I
The main contribution of exp(λ+ (β)) is thus bounded above
by
C0 β exp
max {x(β + M) − xv (x)} [V (2β) − x0 ]
x0 ≤x≤V (2β)
I
The exponent has a unique maximum by strict convexity of
xv (x), and we choose β implicitly by choosing the critical
point to satisfy x = xβ = U = U(ln(2d)). This yields
w (x) = β + M for w (x) = (xv (x))0 ,
so the maximum exponent is written
U(w (U) − v (U)) = U 2 v 0 (U). Lemma 2 follows after some
work to show that under the stated growth condition
ln(βV (2β)) = o(U 2 v 0 (U)).
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
The Poisson case
I
By Stirling’s formula ln(x!) = c + o(1) + (x + 1/2) ln(x) − x,
so we find that the Poisson case is represented by
v (x) = ln(x) − 1 with δ(x) = O(ln(x)). The growth condition
is just satisfied: xv 0 (x) = 1, and the regularity condition
(xv (x))02 /(xv (x))00 ∼ x ln(x)2 is easily satisfied.
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage
Overview
Directed Last Passage Site Model and Directed Polymer Model
Relation between f (β) and νd
Asymptotic Evaluation of νd for Exponential Weights
Asymptotics of νd for a Class of Distributions
The Poisson case
I
By Stirling’s formula ln(x!) = c + o(1) + (x + 1/2) ln(x) − x,
so we find that the Poisson case is represented by
v (x) = ln(x) − 1 with δ(x) = O(ln(x)). The growth condition
is just satisfied: xv 0 (x) = 1, and the regularity condition
(xv (x))02 /(xv (x))00 ∼ x ln(x)2 is easily satisfied.
I
We have U as the inverse of xv (x), or U(u) ∼ u/ ln(u).
Therefore
νd ∼ ln(2d)/ ln ln(2d).
Gregory J. Morrow Department of Mathematics University of Colorado
A directed
at Colorado
polymerSprings
approach to the once-oriented last passage