Directed polymers
RSK correspondence
Geometric RSK
Conclusions
Exact solvability
in directed random polymer models
PhD student: Elia Bisi
Supervisor: Nikos Zygouras
Department of Statistics - University of Warwick
25 June 2015
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Directed polymers
RSK correspondence
Geometric RSK
Conclusions
Summary
1
Directed polymers
2
RSK correspondence
3
Geometric RSK
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Directed polymers
RSK correspondence
Geometric RSK
Conclusions
Polymers
Polymer: large molecule consisting of smaller molecules called
monomers and tied together by chemical bonds.
Linear polymer: linear structure without multiple cross
connections.
Examples of linear polymers: DNA, RNA, proteins.
Figure: A DNA helix.
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Directed polymers
RSK correspondence
Geometric RSK
Conclusions
Directed random polymers in a random potential
Model:
set of allowed paths
Xn = x = (i, xi )ni=0 ∈ (N×Zd )n+1 : x0 = 0, xi−1 ∼ xi ∀i ;
random potential: i.i.d. field on (Ω, F, P, E)
{V (i, x) :
i ∈ N,
x ∈ Zd } ;
Hamiltonian function
Hnβ (x)
= −β
n
X
V (i, xi ) ,
x ∈ Xn .
i=1
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Directed polymers
RSK correspondence
Geometric RSK
Conclusions
Directed random polymers in a random potential
Model:
set of allowed paths
Xn = x = (i, xi )ni=0 ∈ (N×Zd )n+1 : x0 = 0, xi−1 ∼ xi ∀i ;
random potential: i.i.d. field on (Ω, F, P, E)
{V (i, x) :
i ∈ N,
x ∈ Zd } ;
Hamiltonian function
Hnβ (x)
= −β
n
X
V (i, xi ) ,
x ∈ Xn .
i=1
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Directed polymers
RSK correspondence
Geometric RSK
Conclusions
Directed random polymers in a random potential
Model:
set of allowed paths
Xn = x = (i, xi )ni=0 ∈ (N×Zd )n+1 : x0 = 0, xi−1 ∼ xi ∀i ;
random potential: i.i.d. field on (Ω, F, P, E)
{V (i, x) :
i ∈ N,
x ∈ Zd } ;
Hamiltonian function
Hnβ (x)
= −β
n
X
V (i, xi ) ,
x ∈ Xn .
i=1
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Directed polymers
RSK correspondence
Geometric RSK
Conclusions
Polymer measure in 1 + d dimensions
Polymer measure on Xn :
Pnβ (x) =
1
Znβ
β
e −Hn (x) Pn (x) ,
Pn : law of the n-step simple random walk on Zd ;
Znβ : partition sum
n
X
XY
β
β
e βV (i,xi )
Znβ = En e −Hn =
e −Hn (x) Pn (x) =
.
2d
x∈Xn
x∈Xn i=1
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Directed polymers
RSK correspondence
Geometric RSK
Conclusions
Infinite and zero temperature limits
β = 0 (or constant potential): simple random walk on Zd .
β → ∞:
Znβ
X
n
1 X
=
exp β
V (i, xi )
(2d)n
i=1
x∈Xn
n
X
1
exp β max
V (i, xi ) .
x∈Xn
(2d)n
i=1
connection to directed last passage percolation:
n
X
1
log Znβ = max
V (i, xi ) =: Tn
β→∞ β
x∈Xn
lim
i=1
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Directed polymers
RSK correspondence
Geometric RSK
Conclusions
Infinite and zero temperature limits
β = 0 (or constant potential): simple random walk on Zd .
β → ∞:
Znβ
X
n
1 X
=
exp β
V (i, xi )
(2d)n
i=1
x∈Xn
n
X
1
exp β max
V (i, xi ) .
x∈Xn
(2d)n
i=1
connection to directed last passage percolation:
n
X
1
log Znβ = max
V (i, xi ) =: Tn
β→∞ β
x∈Xn
lim
i=1
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Directed polymers
RSK correspondence
Geometric RSK
Conclusions
(Generalized) RSK correspondence
RSK
RSK
Define the map : (R≥0 )n×n −−→ (R≥0 )n×n , W −−→ T :
li,j replaces the submatrix
wi−1,j−1 wi−1,j
wi,j−1
wi,j
with its image under the map
a b
min(b, c) − a
b
→
;
c d
c
max(b, c) + d
(
l1,j−i+1 ◦ · · · ◦ li−1,j−1 ◦ li,j i ≤ j
i
%j :=
li−j+1,1 ◦ · · · ◦ li−1,j−1 ◦ li,j i ≥ j ;
Dk : composition of all %ij such that i + j = k + 1;
RSK: D2n−1 ◦ · · · ◦ D1 .
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Directed polymers
RSK correspondence
Geometric RSK
Conclusions
Properties of RSK
By T = (Z , Z 0 ) we mean:

t1,1 t1,2

 t2,1




tn,1
t1,n
tn−1,n
tn,n−1
tn,n








=


0
zn−1,n−1
λn
0
z1,1

 z
n−1,n−1


0
zn−1,1
z1,1
zn−1,1
λ1




.



Properties:
j
X
zj,i −
i=1
i
X
j=1
j−1
X
zj−1,i =
i=1
0
zi,j
−
i−1
X
j=1
X
wi,j
∀j ;
wi,j
∀i .
i
0
zi−1,j
=
X
j
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Directed polymers
RSK correspondence
Geometric RSK
Conclusions
Properties of RSK
By T = (Z , Z 0 ) we mean:

t1,1 t1,2

 t2,1




tn,1
t1,n
tn−1,n
tn,n−1
tn,n








=


0
zn−1,n−1
λn
0
z1,1

 z
n−1,n−1


0
zn−1,1
z1,1
zn−1,1
λ1




.



Properties:
j
X
zj,i −
i=1
i
X
j=1
j−1
X
zj−1,i =
i=1
0
zi,j
−
i−1
X
j=1
X
wi,j
∀j ;
wi,j
∀i .
i
0
zi−1,j
=
X
j
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Directed polymers
RSK correspondence
Geometric RSK
Conclusions
Last passage percolation
RSK
W −−→ T = (Z , Z 0 );
λ = (λ1 , λ2 , . . . ): shape of Z ,
(n, n)
Z 0;
Πn : directed paths (1, 1) → (n, n);
X
tn,n = λ1 = max
wi,j .
π∈Πn
(1, 1)
(i,j)∈π
tn,n is a point-to-point version of the maximal passage time T2n−2 .
(0, 0)
(2n − 2, 0)
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Directed polymers
RSK correspondence
Geometric RSK
Conclusions
Last passage percolation
RSK
W −−→ T = (Z , Z 0 );
λ = (λ1 , λ2 , . . . ): shape of Z ,
(n, n)
Z 0;
Πn : directed paths (1, 1) → (n, n);
X
tn,n = λ1 = max
wi,j .
π∈Πn
(1, 1)
(i,j)∈π
tn,n is a point-to-point version of the maximal passage time T2n−2 .
(0, 0)
(2n − 2, 0)
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Directed polymers
RSK correspondence
Geometric RSK
Conclusions
Last passage percolation with geometric weights
If wi,j are independent and Geometric(pj qi ):
P(shape = λ) = c · sλ (p1 , . . . , pn ) · sλ (q1 , . . . , qn ) ;
X
P(tn,n ≤ t) = c
sλ (p1 , . . . , pn ) · sλ (q1 , . . . , qn ) .
λ: λ1 ≤t
Here:
c=
n
Y
(1 − pj qi ) ;
i,j=1
sλ is a Schur polynomial.
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Directed polymers
RSK correspondence
Geometric RSK
Conclusions
Geometric RSK
Formally change operations:
a+b → a·b,
a−b →
a
,
b
max(a, b) → a + b .
Now li,j replaces the submatrix
wi−1,j−1 wi−1,j
wi,j−1
wi,j
with its image under the map
a b
bc/[a(b + c)]
b
→
.
c d
c
(b + c)d
gRSK
The new map : (R>0 )n×n −−−→ (R>0 )n×n is called geometric
RSK.
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Directed polymers
RSK correspondence
Geometric RSK
Conclusions
Geometric RSK
Formally change operations:
a+b → a·b,
a−b →
a
,
b
max(a, b) → a + b .
Now li,j replaces the submatrix
wi−1,j−1 wi−1,j
wi,j−1
wi,j
with its image under the map
a b
bc/[a(b + c)]
b
→
.
c d
c
(b + c)d
gRSK
The new map : (R>0 )n×n −−−→ (R>0 )n×n is called geometric
RSK.
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Directed polymers
RSK correspondence
Geometric RSK
Conclusions
Geometric RSK and directed polymers
gRSK
W −−−→ T = (Z , Z 0 );
(n, n)
λ = (λ1 , λ2 , . . . ): shape of Z , Z 0 ;
Πn : directed paths (1, 1) → (n, n);
X Y
tn,n = λ1 =
wi,j .
(1, 1)
π∈Πn (i,j)∈π
β
tn,n is a point-to-point version of the polymer partition sum Z2n−2
.
(0, 0)
(2n − 2, 0)
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Directed polymers
RSK correspondence
Geometric RSK
Conclusions
Geometric RSK and directed polymers
gRSK
W −−−→ T = (Z , Z 0 );
(n, n)
λ = (λ1 , λ2 , . . . ): shape of Z , Z 0 ;
Πn : directed paths (1, 1) → (n, n);
X Y
tn,n = λ1 =
wi,j .
(1, 1)
π∈Πn (i,j)∈π
β
tn,n is a point-to-point version of the polymer partition sum Z2n−2
.
(0, 0)
(2n − 2, 0)
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Directed polymers
RSK correspondence
Geometric RSK
Conclusions
Properties of geometric RSK
n
Y
Qj
n
Y
Qj
i=1 zj,i
wi,j = Qj−1
i=1 zj−1,i
i=1
j=1
0
i=1 zj,i
0
i=1 zj−1,i
wi,j = Qj−1
X 1
1
=
+
wi,j
t1,1
i,j
X
(i,j)6=(1,1)
∀j ;
∀i ;
ti−1,j + ti,j−1
ti,j
log(wi,j )1≤i,j≤n → log(ti,j )1≤i,j≤n has Jacobian ±1.
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Directed polymers
RSK correspondence
Geometric RSK
Conclusions
Inverse gamma polymer
Theorem (Corwin, O’Connell, Seppäläinen, Zygouras 2014)
−1
If wi,j are independent and wi,j
∼ Γ(αj + α̂i , 1), the Laplace
transform of tn,n is
Y
E e −θtn,n =
i,j
1
Γ(αj + α̂i )
Z
−1
e −θλ1 −λn ψα (λ)ψα̂ (λ)
(R>0 )n
n
Y
dλi
i=1
λi
.
Here, ψα and ψα̂ are Whittaker functions:
−αj
Z Y
n Qj
i=1 zj,i
exp
ψα (λ) =
Qj−1
i=1 zj−1,i
j=1
X zi−1,j + zi+1,j+1
−
zi,j
(i,j)
!
Y
1≤j≤i<n
dzi,j
,
zi,j
where the integral is over all triangular patterns (zi,j ) of height n
and fixed shape λ = (zn,1 , . . . , zn,n ).
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Directed polymers
RSK correspondence
Geometric RSK
Conclusions
Variations and open problems
Variations on the polymer model with inverse gamma weights
(O’Connell, Seppäläinen and Zygouras 2014):
symmetrized polymers;
polymers forced to stay below a hard wall.
Open problems:
antisymmetrized polymers;
polymers not allowed to go outside a diagonal strip.
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Directed polymers
RSK correspondence
Geometric RSK
Conclusions
Variations and open problems
Variations on the polymer model with inverse gamma weights
(O’Connell, Seppäläinen and Zygouras 2014):
symmetrized polymers;
polymers forced to stay below a hard wall.
Open problems:
antisymmetrized polymers;
polymers not allowed to go outside a diagonal strip.
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Directed polymers
RSK correspondence
Geometric RSK
Conclusions
Recap
We have seen:
(1 + d)-dim. directed random polymers in random
environment;
RSK correspondence
(1 + 1)-dim. last passage percolation and the exactly solvable
case of geometric weights;
geometric RSK correspondence
(1 + 1)-dim. polymer partition sum and the exactly solvable
case of inverse gamma weights.
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