Large deviations for certain inhomogeneous corner
growth models
Chris Janjigian
University of Wisconsin-Madison
May 2016
(joint work with Elnur Emrah)
Chris Janjigian
Large deviations for certain inhomogeneous corner growth models
1/24
Corner growth model
j
1
1
Chris Janjigian
i
Large deviations for certain inhomogeneous corner growth models
2/24
Corner growth model
j
1
1
i
Take W pi, jq, pi, jq P N2 .
Chris Janjigian
Large deviations for certain inhomogeneous corner growth models
2/24
Corner growth model
j
1
1
i
Take W pi, jq, pi, jq P N2 . pm, nq enters at time G pm, nq:
G pm, nq “ G pm ´ 1, nq _ G pm, n ´ 1q ` W pm, nq
G pm, 0q “ G p0, nq “ 0
Chris Janjigian
Large deviations for certain inhomogeneous corner growth models
2/24
Corner growth model
j
D
1
(
1
i
Take W pi, jq, pi, jq P N2 . pm, nq enters at time G pm, nq:
G pm, nq “ G pm ´ 1, nq _ G pm, n ´ 1q ` W pm, nq
G pm, 0q “ G p0, nq “ 0
Chris Janjigian
Large deviations for certain inhomogeneous corner growth models
2/24
Corner growth model
j
D
1
1
i
Take W pi, jq, pi, jq P N2 . pm, nq enters at time G pm, nq:
G pm, nq “ G pm ´ 1, nq _ G pm, n ´ 1q ` W pm, nq
G pm, 0q “ G p0, nq “ 0
Chris Janjigian
Large deviations for certain inhomogeneous corner growth models
2/24
Corner growth model
j
+
1
ˆ
1
i
Take W pi, jq, pi, jq P N2 . pm, nq enters at time G pm, nq:
G pm, nq “ G pm ´ 1, nq _ G pm, n ´ 1q ` W pm, nq
G pm, 0q “ G p0, nq “ 0
Chris Janjigian
Large deviations for certain inhomogeneous corner growth models
2/24
Corner growth model
j
+
ˆ
1
1
i
Take W pi, jq, pi, jq P N2 . pm, nq enters at time G pm, nq:
G pm, nq “ G pm ´ 1, nq _ G pm, n ´ 1q ` W pm, nq
G pm, 0q “ G p0, nq “ 0
Chris Janjigian
Large deviations for certain inhomogeneous corner growth models
2/24
Corner growth model
j
+

1
1
i
Take W pi, jq, pi, jq P N2 . pm, nq enters at time G pm, nq:
G pm, nq “ G pm ´ 1, nq _ G pm, n ´ 1q ` W pm, nq
G pm, 0q “ G p0, nq “ 0
Chris Janjigian
Large deviations for certain inhomogeneous corner growth models
2/24
Corner growth model
j
+


1
1
i
Take W pi, jq, pi, jq P N2 . pm, nq enters at time G pm, nq:
G pm, nq “ G pm ´ 1, nq _ G pm, n ´ 1q ` W pm, nq
G pm, 0q “ G p0, nq “ 0
Chris Janjigian
Large deviations for certain inhomogeneous corner growth models
2/24
Corner growth model - LPP
j
n
1
i
m
1
By induction,
G pm, nq “
Chris Janjigian
max
ÿ
up-right paths
π:p1,1qÑpm,nq pi,jqPπ
W pi, jq.
Large deviations for certain inhomogeneous corner growth models
3/24
Corner growth model - LPP
j
n
1
i
m
1
By induction,
G pm, nq “
Chris Janjigian
max
ÿ
up-right paths
π:p1,1qÑpm,nq pi,jqPπ
W pi, jq.
Large deviations for certain inhomogeneous corner growth models
3/24
Corner growth model - LPP
j
n
1
i
m
1
By induction,
G pm, nq “
Chris Janjigian
max
ÿ
up-right paths
π:p1,1qÑpm,nq pi,jqPπ
W pi, jq.
Large deviations for certain inhomogeneous corner growth models
3/24
Corner growth model - LPP
j
n
1
i
m
1
By induction,
G pm, nq “
Chris Janjigian
max
ÿ
up-right paths
π:p1,1qÑpm,nq pi,jqPπ
W pi, jq.
Large deviations for certain inhomogeneous corner growth models
3/24
Corner growth model - exclusion
j
i
x
pm, nq enters when particle n (count right-to-left) interchanges with
hole m (count left-to-right).
Chris Janjigian
Large deviations for certain inhomogeneous corner growth models
4/24
Corner growth model - exclusion
j
i
x
pm, nq enters when particle n (count right-to-left) interchanges with
hole m (count left-to-right).
Chris Janjigian
Large deviations for certain inhomogeneous corner growth models
4/24
Corner growth model - exclusion
j
D
(
D (
i
x
pm, nq enters when particle n (count right-to-left) interchanges with
hole m (count left-to-right).
Chris Janjigian
Large deviations for certain inhomogeneous corner growth models
4/24
Corner growth model - exclusion
j
D
D
i
x
pm, nq enters when particle n (count right-to-left) interchanges with
hole m (count left-to-right).
Chris Janjigian
Large deviations for certain inhomogeneous corner growth models
4/24
Corner growth model - exclusion
j
+
ˆ
+ ˆ i
x
pm, nq enters when particle n (count right-to-left) interchanges with
hole m (count left-to-right).
Chris Janjigian
Large deviations for certain inhomogeneous corner growth models
4/24
Corner growth model - exclusion
j
+
ˆ
+
ˆ i
x
pm, nq enters when particle n (count right-to-left) interchanges with
hole m (count left-to-right).
Chris Janjigian
Large deviations for certain inhomogeneous corner growth models
4/24
Corner growth model - exclusion
j
+
+ 

i
x
pm, nq enters when particle n (count right-to-left) interchanges with
hole m (count left-to-right).
Chris Janjigian
Large deviations for certain inhomogeneous corner growth models
4/24
Corner growth model - exclusion
j
+


+   i
x
pm, nq enters when particle n (count right-to-left) interchanges with
hole m (count left-to-right).
Chris Janjigian
Large deviations for certain inhomogeneous corner growth models
4/24
Shape theorem
Under mild assumptions,
lim n´1 G pt ns u, t nt uq “ g ps, tq (a.s.)
nÑ8
Chris Janjigian
Large deviations for certain inhomogeneous corner growth models
5/24
Solvable corner growth models
W pi, jq
j
i
Chris Janjigian
Large deviations for certain inhomogeneous corner growth models
6/24
Solvable corner growth models
i.i.d.
W pi, jq „
Exppcq
j
Homogeneous:
i.i.d.
W pi, jq „ Exppcq
(or Geoppq).
i
Chris Janjigian
Large deviations for certain inhomogeneous corner growth models
6/24
Solvable corner growth models
ind.
W pi, jq „
Exppai ` bj q
j
Homogeneous:
i.i.d.
W pi, jq „ Exppcq
(or Geoppq).
Inhomogeneous:
ind.
W pi, jq „ Exppai ` bj q
a “ pan qně1 and b “ pbn qně1
ai , bj ě c ą 0.
i
Chris Janjigian
Large deviations for certain inhomogeneous corner growth models
6/24
Stationary model
Extend environment to include pi, 0q, p0, jq, i, j ě 0, W p0, 0q “ 0.
j
X p1,
p0, 0q
1q
1 Y
1
Chris Janjigian
i
Large deviations for certain inhomogeneous corner growth models
7/24
Stationary model
Extend environment to include pi, 0q, p0, jq, i, j ě 0, W p0, 0q “ 0.
j
1
X p1,
Y
p0, 0q
1q
1
Chris Janjigian
i
Large deviations for certain inhomogeneous corner growth models
7/24
Stationary model
Extend environment to include pi, 0q, p0, jq, i, j ě 0, W p0, 0q “ 0.
j
Take z: ´ai ă z ă bj , i, j P N
W pi, 0q „ Exppai ` zq
X p1,
p0, 0q
1q
1 Y
1 Chris Janjigian
" '
i
Large deviations for certain inhomogeneous corner growth models
7/24
Stationary model
Extend environment to include pi, 0q, p0, jq, i, j ě 0, W p0, 0q “ 0.
j
F
:
.
"
1
Take z: ´ai ă z ă bj , i, j P N
W pi, 0q „ Exppai ` zq
W p0, jq „ Exppbj ´ zq
X p1,
Y
p0, 0q
1q
1
Chris Janjigian
i
Large deviations for certain inhomogeneous corner growth models
7/24
Stationary model
Extend environment to include pi, 0q, p0, jq, i, j ě 0, W p0, 0q “ 0.
j
1
Xp1,
Y
p0, 0q
1q
1
Chris Janjigian
Take z: ´ai ă z ă bj , i, j P N
W pi, 0q „ Exppai ` zq
W p0, jq „ Exppbj ´ zq
W pi, jq „ Exppai ` bj q
i
Large deviations for certain inhomogeneous corner growth models
7/24
Stationary model
Extend environment to include pi, 0q, p0, jq, i, j ě 0, W p0, 0q “ 0.
j
Take z: ´ai ă z ă bj , i, j P N
W pi, 0q „ Exppai ` zq
W p0, jq „ Exppbj ´ zq
W pi, jq „ Exppai ` bj q
pi, jq enters at time Ĝ pi, jq
X p1,
p0, 0q
1q
1 Y
1
Chris Janjigian
i
Large deviations for certain inhomogeneous corner growth models
7/24
Stationary model
Extend environment to include pi, 0q, p0, jq, i, j ě 0, W p0, 0q “ 0.
j
Take z: ´ai ă z ă bj , i, j P N
W pi, 0q „ Exppai ` zq
W p0, jq „ Exppbj ´ zq
W pi, jq „ Exppai ` bj q
pi, jq enters at time Ĝ pi, jq
X pi, jq “ Ĝ pi, jq ´ Ĝ pi ´ 1, jq
X p1,
p0, 0q
1q
1 Y
1
Chris Janjigian
i
Large deviations for certain inhomogeneous corner growth models
7/24
Stationary model
Extend environment to include pi, 0q, p0, jq, i, j ě 0, W p0, 0q “ 0.
j
Take z: ´ai ă z ă bj , i, j P N
W pi, 0q „ Exppai ` zq
W p0, jq „ Exppbj ´ zq
W pi, jq „ Exppai ` bj q
pi, jq enters at time Ĝ pi, jq
X pi, jq “ Ĝ pi, jq ´ Ĝ pi ´ 1, jq
X p1,
p0, 0q
1q
1 Y
1
Chris Janjigian
i
Y pi, jq “ Ĝ pi, jq ´ Ĝ pi, j ´ 1q
Large deviations for certain inhomogeneous corner growth models
7/24
Stationary model
X pi, jq „ Exppai ` zq
W pi, jq „ Exppai ` bj q
Y pi ´ 1, jq „ Exppbj ´ zq
Y pi, jq „ Exppbj ´ zq
X pi, j ´ 1q „ Exppai ` zq
Chris Janjigian
Large deviations for certain inhomogeneous corner growth models
8/24
Stationary model
X pi, jq „ Exppai ` zq
W pi, jq „ Exppai ` bj q
Y pi ´ 1, jq „ Exppbj ´ zq
Y pi, jq „ Exppbj ´ zq
X pi, j ´ 1q „ Exppai ` zq
X pi, jq “ X pi, j ´ 1q ´ X pi, j ´ 1q ^ Y pi ´ 1, jq ` W pi, jq
Y pi, jq “ Y pi ´ 1, jq ´ X pi, j ´ 1q ^ Y pi ´ 1, jq ` W pi, jq
Chris Janjigian
Large deviations for certain inhomogeneous corner growth models
8/24
Stationary model
X pi, jq „ Exppai ` zq
W pi, jq „ Exppai ` bj q
Y pi ´ 1, jq „ Exppbj ´ zq
Y pi, jq „ Exppbj ´ zq
X pi, j ´ 1q „ Exppai ` zq
X pi, jq “ X pi, j ´ 1q ´ X pi, j ´ 1q ^ Y pi ´ 1, jq ` W pi, jq
Y pi, jq “ Y pi ´ 1, jq ´ X pi, j ´ 1q ^ Y pi ´ 1, jq ` W pi, jq
Key point: This map preserves the joint distribution of pX , Y q.
Chris Janjigian
Large deviations for certain inhomogeneous corner growth models
8/24
Stationary model
j
X pi, 0q “ W pi, 0q „ Exppai ` zq
Y p0, jq “ W p0, jq „ Exppbj ´ zq
i
Chris Janjigian
Large deviations for certain inhomogeneous corner growth models
9/24
Stationary model
j
X pi, 0q “ W pi, 0q „ Exppai ` zq
Y p0, jq “ W p0, jq „ Exppbj ´ zq
By induction, X pi, jq, Y pi, jq
are mut. indep. along
down-right paths
X pi, jq „ Exppai ` zq
Y pi, jq „ Exppbj ´ zq.
i
Chris Janjigian
(Burke property)
Large deviations for certain inhomogeneous corner growth models
9/24
Stationary model
j
X pi, 0q “ W pi, 0q „ Exppai ` zq
Y p0, jq “ W p0, jq „ Exppbj ´ zq
By induction, X pi, jq, Y pi, jq
are mut. indep. along
down-right paths
X pi, jq „ Exppai ` zq
Y pi, jq „ Exppbj ´ zq.
i
Chris Janjigian
(Burke property)
Large deviations for certain inhomogeneous corner growth models
9/24
Stationary model
j
X pi, 0q “ W pi, 0q „ Exppai ` zq
Y p0, jq “ W p0, jq „ Exppbj ´ zq
By induction, X pi, jq, Y pi, jq
are mut. indep. along
down-right paths
X pi, jq „ Exppai ` zq
Y pi, jq „ Exppbj ´ zq.
i
Chris Janjigian
(Burke property)
Large deviations for certain inhomogeneous corner growth models
9/24
Stationary model
j
X pi, 0q “ W pi, 0q „ Exppai ` zq
Y p0, jq “ W p0, jq „ Exppbj ´ zq
By induction, X pi, jq, Y pi, jq
are mut. indep. along
down-right paths
X pi, jq „ Exppai ` zq
Y pi, jq „ Exppbj ´ zq.
i
Chris Janjigian
(Burke property)
Large deviations for certain inhomogeneous corner growth models
9/24
Random environment
a “ pai q, b “ pbj q indep. i.i.d., finite
ind.
W pi, jq „
Exppai ` bj q mean ě c ą 0 sequences (can be
weaker)
..
.
bj
..
.
b1
a1
Chris Janjigian
...
ai
...
Large deviations for certain inhomogeneous corner growth models
10/24
Random environment
a “ pai q, b “ pbj q indep. i.i.d., finite
ind.
W pi, jq „
Exppai ` bj q mean ě c ą 0 sequences (can be
weaker)
..
.
bj
Pa,b : conditioned on pa, bq,
..
.
ind.
W pi, jq „ Exppai ` bj q.
b1
a1
Chris Janjigian
...
ai
...
Large deviations for certain inhomogeneous corner growth models
10/24
Random environment
a “ pai q, b “ pbj q indep. i.i.d., finite
ind.
W pi, jq „
Exppai ` bj q mean ě c ą 0 sequences (can be
weaker)
..
.
bj
Pa,b : conditioned on pa, bq,
..
.
ind.
W pi, jq „ Exppai ` bj q.
b1
P: average Pa,b over pa, bq.
a1
Chris Janjigian
...
ai
...
Large deviations for certain inhomogeneous corner growth models
10/24
Random environment
Key points:
..
.
ind.
W pi, jq „
Exppai ` bj q
bj
..
.
b1
a1
Chris Janjigian
...
ai
...
Large deviations for certain inhomogeneous corner growth models
11/24
Random environment
Key points:
..
.
ind.
W pi, jq „
Exppai ` bj q
bj
Pa,b : indep., not ident. dist.:
if i ‰ i 1 or j ‰ j 1
d
W pi, jq ‰ W pi 1 , j 1 q (usually)
W pi, jq K W pi 1 , j 1 q
..
.
b1
a1
Chris Janjigian
...
ai
...
Large deviations for certain inhomogeneous corner growth models
11/24
Random environment
Key points:
..
.
ind.
W pi, jq „
Exppai ` bj q
bj
Pa,b : indep., not ident. dist.:
if i ‰ i 1 or j ‰ j 1
d
W pi, jq ‰ W pi 1 , j 1 q (usually)
W pi, jq K W pi 1 , j 1 q
..
.
P: ident. dist., not indep.:
d
b1
a1
Chris Janjigian
...
ai
...
W pi, jq “ W pi 1 , j 1 q
if i “ i 1 or j “ j 1
CovpW pi, jq, W pi 1 , j 1 qq ‰ 0.
Large deviations for certain inhomogeneous corner growth models
11/24
Random environment
Key points:
Pa,b : indep., not ident. dist.:
if i ‰ i 1 or j ‰ j 1
..
.
d
W pi, jq ‰ W pi 1 , j 1 q (usually)
W pi, jq K W pi 1 , j 1 q
bj
..
.
P: ident. dist., not indep.:
d
b1
a1
Chris Janjigian
...
ai
...
W pi, jq “ W pi 1 , j 1 q
if i “ i 1 or j “ j 1
CovpW pi, jq, W pi 1 , j 1 qq ‰ 0.
Large deviations for certain inhomogeneous corner growth models
11/24
Shape function
Theorem (Emrah ’15)
For s, t ą 0, P almost surely and for almost all pa, bq Pa,b almost surely
" „

„
*
1
1
sE
lim n´1 G pt ns u, t nt uq “ min
`tE
nÑ8
´αďzďβ
a1 ` z
b1 ´ z
¯
¯
Key points:
Only depends on marginal distributions of a1 and b1 separately.
Notation: α “ essinfta1 u, β “ essinftb1 u.
¯
¯
Tractable 1D minmization problem.
“
‰
“
‰
gz ps, tq “ s E pa1 ` zq´1 ` t E pb1 ´ zq´1 is the shape function
in the stationary version of the model.
Chris Janjigian
Large deviations for certain inhomogeneous corner growth models
12/24
Asymptotic shape of the cluster
t
S1
g is strictly concave in S, linear
in S1 and S2 .
S1 , S2 ‰ H iff
S
E rpa1 ´ αq´2 s ă 8 pS1 q
¯
E rpb1 ´ βq´2 s ă 8 pS2 q
¯
g ď1
S2
0
Chris Janjigian
s
Large deviations for certain inhomogeneous corner growth models
13/24
Right tail large deviations
Pick a direction ps, tq. Interested in
t
l1
S1
S
l2
g ps, tq “ 1
S2
0
Chris Janjigian
s
Large deviations for certain inhomogeneous corner growth models
14/24
Right tail large deviations
Pick a direction ps, tq. Interested in
t
l1
r ą g ps, tq
S1
Ppn´1 G pt ns u, t nt uq ě r q « e ´n Js,t pr q
Ppn´1 G pt ns u, t nt uq ě r q « e ´n Js,t pr q
S
l2
g ps, tq “ 1
S2
0
Chris Janjigian
s
Large deviations for certain inhomogeneous corner growth models
14/24
Right tail large deviations
Pick a direction ps, tq. Interested in
t
l1
r ą g ps, tq
S1
Ppn´1 G pt ns u, t nt uq ě r q « e ´n Js,t pr q
Ppn´1 G pt ns u, t nt uq ě r q « e ´n Js,t pr q
S
Is there a difference between
ps, tq P S and ps, tq P S1 , S2 ?
What happens when
ps, tq P l1 , l2 ?
l2
g ps, tq “ 1
S2
0
Chris Janjigian
s
Large deviations for certain inhomogeneous corner growth models
14/24
Right tail large deviations
Pick a direction ps, tq. Interested in
t
l1
r ą g ps, tq
S1
Ppn´1 G pt ns u, t nt uq ě r q « e ´n Js,t pr q
Ppn´1 G pt ns u, t nt uq ě r q « e ´n Js,t pr q
S
Is there a difference between
ps, tq P S and ps, tq P S1 , S2 ?
What happens when
ps, tq P l1 , l2 ?
l2
g ps, tq “ 1
S2
0
Chris Janjigian
Is there a relationship between
the Js,t pr q and Js,t pr q?
s
Large deviations for certain inhomogeneous corner growth models
14/24
Quenched right tail rate function
Theorem
For almost all pa, bq, for any s, t ą 0 and r ě g ps, tq
`
˘
Js,t pr q “ lim ´n´1 log Pa,b n´1 G pt ns u, t nt uq ě r
nÑ8
˙
ˆ
˙*
"
ˆ
b1 ´ z
a1 ` z ` λ
´ t E log
“
sup
r λ ´ s E log
a1 ` z
b1 ´ z ´ λ
λPp0,α`βq
¯ ¯
zPp´α,β´λq
¯ ¯
Key points:
Only depends on marginal distributions of a1 and b1 separately.
Tractable 2D maximization problem.
LDP with rate function Is,t pr q “ Js,t pr q1tr ěg ps,tqu ` 81tr ăg ps,tqu .
(Left tail rate should be n2 under mild conditions).
Chris Janjigian
Large deviations for certain inhomogeneous corner growth models
15/24
Example rate functions
?
?
If a1 , b1 „ δ c2 then for r ě g ps, tq “ c ´1 p s ` tq2 we have
Js,t pr q “
ˆ
˙
ˆ
˙
a
s ´ t ` cr
t ´ s ` cr
´1
´1
2
?
?
ps ` t ´ cr q ´ 4st ´ 2s cosh
´ 2t cosh
2 csr
2 ctr
which recovers a result of Seppäläinen ’98.
Chris Janjigian
Large deviations for certain inhomogeneous corner growth models
16/24
Example rate functions
If a1 , b1 „ Unifrc{2, c{2 ` ls, then for r ě g ps, sq “
Js,s pr q “ r λ‹ ´
2s
l
ż c{2`l
ˆ
log
c{2
2s
l
`
log 1 `
x ` z‹ ` λ ‹
x ` z‹
2l
c
˘
˙
dx
where
d
z‹ “ ´
pc{2 ` lq2 ´ c 2 e rl{s {4
1 ´ e rl{s
d
z‹ ` λ ‹ “
Chris Janjigian
pc{2 ` lq2 ´ c 2 e rl{s {4
.
1 ´ e rl{s
Large deviations for certain inhomogeneous corner growth models
17/24
Example rate functions
If a1 , b1 „ pδc ``qδd , 0 ď p ď ˘1, q “ 1 ´ p, then for
r ě g ps, sq “ 2s pc ´1 ` qd ´1 ,
˙
ˆ
˙
ˆ
c ´ z‹
c ` z‹ ` λ‹
´ tq log
Js,s pr q “ r λ‹ ´sp log
c ` z‹
c ´ z‹ ´ λ‹
ˆ
˙
ˆ
˙
d ` z‹ ` λ‹
d ´ z‹
´ sq log
´ tq log
d ` z‹
d ´ z‹ ´ λ‹
where
?
2cp ` 2dq ` c 2 r ` d 2 r ´ ∆
2r
?
2cp ` 2dq ` c 2 r ` d 2 r ` ∆
z‹ ` λ‹ “
,
2r
∆ “ p2cp ` 2dq ` c 2 r ` d 2 r q2 ` 4r p2cd 2 p ` 2c 2 dq ´ c 2 d 2 r q.
z‹ “
Chris Janjigian
Large deviations for certain inhomogeneous corner growth models
18/24
Expected fluctuations
t
l1
S1
Quenched fluct. should be
TWGUE in S, but not in S1 , S2 .
S
l2
S2
0
Chris Janjigian
s
Large deviations for certain inhomogeneous corner growth models
19/24
Expected fluctuations
t
l1
S1
Quenched fluct. should be
TWGUE in S, but not in S1 , S2 .
S
Q1: Can we “see” different
scaling exponents in the rate
functions?
l2
S2
0
Chris Janjigian
s
Large deviations for certain inhomogeneous corner growth models
19/24
Expected fluctuations
t
l1
S1
Quenched fluct. should be
TWGUE in S, but not in S1 , S2 .
S
Q1: Can we “see” different
scaling exponents in the rate
functions?
l2
Q2: What happens when
ps, tq P l1 , l2 ?
S2
0
Chris Janjigian
s
Large deviations for certain inhomogeneous corner growth models
19/24
Scaling and the quenched rate functions
Notation: ζ P r´α, βs solves (uniquely) gζ ps, tq “ g ps, tq
¯ ¯
Proposition
For any s, t ą 0, as Ó 0, Js,t pg ps, tq ` q “
$ˆ

„
˙´1
„
2
2
’
’
ps, tq P S1
`tE
2 ` op2 q
’ ´s E
’
’
pa1 ´ αq2
pb1 ` αq2
’
’ ˆ „

„
˙´1{2
¯
¯
’
’
2
1
1
’
’
sE
`tE
3{2 ` op3{2 q ps, tq P l1
’
’
3
3
’
3
pa
´
αq
pb
`
αq
1
1
’
’
„
¯ 
¯ ˙´1{2
&4 ˆ „
1
1
`tE
sE
3{2 ` op3{2 q ps, tq P S
’3
pa1 ` ζq3
pb1 ´ ζq3
’
’
ˆ „

„
˙´1{2
’
’
1
1
2
’
’
sE
`tE
3{2 ` op3{2 q ps, tq P l2
’
’
’
3
pa1 ` βq3
pb1 ´ βq3
’
’
ˆ „

„
˙´1
’
’
2
2 ¯
’
’
s
E
´
t
E
2 ` op2 q
ps, tq P S2
%
pa1 ` βq2
pb1 ´ βq2
¯
¯
Chris Janjigian
Large deviations for certain inhomogeneous corner growth models
20/24
Scaling and the quenched rate functions
Notation: ζ P r´α, βs solves (uniquely) gζ ps, tq “ g ps, tq
¯ ¯
Proposition
For any s, t ą 0, as Ó 0, Js,t pg ps, tq ` q “
$ˆ

„
˙´1
„
2
2
’
’
ps, tq P S1
`tE
2 ` op2 q
’ ´s E
’
’
pa1 ´ αq2
pb1 ` αq2
’
’ ˆ „

„
˙´1{2
¯
¯
’
’
2
1
1
’
’
sE
`tE
3{2 ` op3{2 q ps, tq P l1
’
’
3
3
’
3
pa
´
αq
pb
`
αq
1
1
’
’
„
¯ 
¯ ˙´1{2
&4 ˆ „
1
1
`tE
sE
3{2 ` op3{2 q ps, tq P S
’3
pa1 ` ζq3
pb1 ´ ζq3
’
’
ˆ „

„
˙´1{2
’
’
1
1
2
’
’
sE
`tE
3{2 ` op3{2 q ps, tq P l2
’
’
’
3
pa1 ` βq3
pb1 ´ βq3
’
’
ˆ „

„
˙´1
’
’
2
2 ¯
’
’
s
E
´
t
E
2 ` op2 q
ps, tq P S2
%
pa1 ` βq2
pb1 ´ βq2
¯
¯
Chris Janjigian
Large deviations for certain inhomogeneous corner growth models
20/24
Scaling and the quenched rate functions
Heuristically (not rigorous) consistent with expected TWGUE fluct. in S:
Chris Janjigian
Large deviations for certain inhomogeneous corner growth models
21/24
Scaling and the quenched rate functions
Heuristically (not rigorous) consistent with expected TWGUE fluct. in S:
Take ps, tq P S and set
„

„

ˇ
1
1
1
C “sE
`tE
“ Bz2 gz ps, tqˇz“ζ
3
3
pa ` ζq
pb ´ ζq
2
Chris Janjigian
Large deviations for certain inhomogeneous corner growth models
21/24
Scaling and the quenched rate functions
Heuristically (not rigorous) consistent with expected TWGUE fluct. in S:
Take ps, tq P S and set
„

„

ˇ
1
1
1
C “sE
`tE
“ Bz2 gz ps, tqˇz“ζ
3
3
pa ` ζq
pb ´ ζq
2
2
For n large and large enough r (but not O(n 3 )), we might expect
1
1
Pa,b pG pt ns u, t nt uq ´ ng ps, tq ě n 3 C 3 r q
Chris Janjigian
Large deviations for certain inhomogeneous corner growth models
21/24
Scaling and the quenched rate functions
Heuristically (not rigorous) consistent with expected TWGUE fluct. in S:
Take ps, tq P S and set
„

„

ˇ
1
1
1
C “sE
`tE
“ Bz2 gz ps, tqˇz“ζ
3
3
pa ` ζq
pb ´ ζq
2
2
For n large and large enough r (but not O(n 3 )), we might expect
1
1
1
Pa,b pG pt ns u, t nt uq ´ ng ps, tq ě n 3 C 3 r q « e ´n Js,t pC 3 n
Chris Janjigian
´2
3
Large deviations for certain inhomogeneous corner growth models
rq
21/24
Scaling and the quenched rate functions
Heuristically (not rigorous) consistent with expected TWGUE fluct. in S:
Take ps, tq P S and set
„

„

ˇ
1
1
1
C “sE
`tE
“ Bz2 gz ps, tqˇz“ζ
3
3
pa ` ζq
pb ´ ζq
2
2
For n large and large enough r (but not O(n 3 )), we might expect
1
1
1
Pa,b pG pt ns u, t nt uq ´ ng ps, tq ě n 3 C 3 r q « e ´n Js,t pC 3 n
4
« e´ 3 C
´1
2
´2
3
1
rq
2
3
pC 3 n´ 3 r q 2 n
4
3
“ e´ 3 r 2
which agrees with the leading order TWGUE right tail.
Chris Janjigian
Large deviations for certain inhomogeneous corner growth models
21/24
Annealed large deviations
Theorem
For s, t ą 0 and r ě g ps, tq,
`
˘
Js,t pr q “ lim ´n´1 log P n´1 G pt ns u, t nt uq ě r
nÑ8
"
„

„
*
a1 ` z ` λ
b1 ´ z
“
sup
r λ ´ s log E
´ t log E
a1 ` z
b1 ´ z ´ λ
λPp0,α`βq
¯ ¯
zPp´α,β´λq
¯ ¯
Key points
Very similar to quenched rate function - only difference is E Ø log
Variational problem still tractable (though no explicit examples)
For all s, t ą 0, Js,t pg ps, tq ` q “ Cs,t 2 ` op2 q (Cs,t explicit)
There are rate n left tail pn´1 G pt ns u, t nt uqq ă g ps, tq ´ q
annealed large deviations.
Chris Janjigian
Large deviations for certain inhomogeneous corner growth models
22/24
Variational connection for right tail
Theorem
For any s, t ą 0 and r ě g ps, tq,
Js,t pr q “ inf
ν1 ,ν2
(
Iνs,t1 ,ν2 pr q ` s Hpν1 |αq ` t Hpν2 |βq .
There exists a unique (explicit) minimizing pair pν1 , ν2 q for which equality
holds.
Key points:
Notation: α is the dist. of a1 , β is the dist of b1 , Hp¨|¨q is relative
entropy.
Iνs,t1 ,ν2 pr q is quenched rate function for the model with marginals
a1 „ ν1 , b1 „ ν2 .
Chris Janjigian
Large deviations for certain inhomogeneous corner growth models
23/24
Thanks!
Chris Janjigian
Large deviations for certain inhomogeneous corner growth models
24/24