Large deviations for certain inhomogeneous corner growth ... · Outline 1 The (totally asymmetric)...
Transcript of Large deviations for certain inhomogeneous corner growth ... · Outline 1 The (totally asymmetric)...
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Large deviations for certain inhomogeneous cornergrowth models
Chris Janjigian
Universite Paris Diderot
April 28, 2017(joint work with Elnur Emrah)
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Outline
1 The (totally asymmetric) corner growth model1 Questions: general homogeneous shape theorem and large
deviations2 Inhomogeneous exponential model3 Model with random parameters
2 Results and proof sketches1 The shape function: appearance of linear regions.2 Overview of quenched and annealed large deviation results.3 Sketch of quenched rate function computation.
Goal: To better understand what can happen in inhomogeneousmodels in the KPZ class at the level of large deviations through asolvable example.
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(Homogeneous) Last passage percolation
j
i1
1 m
n
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Take W pi , jq ě 0, pi , jq P N2 i.i.d.
G pm, nq “ maxup-right pathsπ:p1,1qÑpm,nq
ÿ
pi,jqPπ
W pi , jq
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(Homogeneous) Last passage percolation
j
i1
1 m
n
Take W pi , jq ě 0, pi , jq P N2 i.i.d.
G pm, nq “ maxup-right pathsπ:p1,1qÑpm,nq
ÿ
pi,jqPπ
W pi , jq
Chris Janjigian Large deviations for certain inhomogeneous corner growth models 3/29
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Homogeneous shape theorem
Theorem (Martin, ’04)
Suppose the family tW pi , jqu are positive, i.i.d. random variables with
ż 8
0
a
PpW p1, 1q ą rqdr ă 8.
Then there exists a finite, concave, homogeneous functiong : p0,8q2 Ñ R` such that
limnÑ8
n´1G pt ns u, t nt uq “ gps, tq.
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Homogeneous right tail large deviations
Theorem
Suppose the family tW pi , jqu are positive, i.i.d. random variables withE reλW p1,1qs ă 8 for some λ ą 0 and that PpW p1, 1q ą rq ą 0 for allr ą 0. Then there exists a finite, convex function Js,tprq : RÑ R` suchthat
limnÑ8
´n´1 logP pG pt ns u, t nt uq ě nrq “ Js,tprq.
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Homogeneous left tail large deviations
Theorem
Suppose the family tW pi , jqu are non-negative, non-degenerate,i.i.d. random variables with E reλW p1,1qs ă 8 for some λ ą 0. Then thereexist constants C ą 0 such that for r P p0, gps, tqq
P pG pt ns u, t nt uq ď nrq ď C´1e´Cn2
.
A lower bound with the same rate holds for all r P p0, gps, tqq ifPpW p1, 1q P r0, εqq ą 0 for all ε ą 0.
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Inhomogeneous exponential last passage percolation
W pi , jq
W pi , jq ind.„ Exppai ` bjq
�j
j
i
i
W pi , jqind.„ Exppai ` bjq
a “ panqně1 and b “ pbnqně1
ai , bj ě c ą 0.
Model introduced byJohansson ’01.
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Inhomogeneous exponential last passage percolation
W pi , jq
W pi , jq ind.„ Exppai ` bjq
�
j
j
i
i
W pi , jqind.„ Exppai ` bjq
a “ panqně1 and b “ pbnqně1
ai , bj ě c ą 0.
Model introduced byJohansson ’01.
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Inhomogeneous exponential last passage percolation
W pi , jq
W pi , jq ind.„ Exppai ` bjq
�
j
j
i
i
W pi , jqind.„ Exppai ` bjq
a “ panqně1 and b “ pbnqně1
ai , bj ě c ą 0.
Model introduced byJohansson ’01.
Chris Janjigian Large deviations for certain inhomogeneous corner growth models 7/29
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Stationary model
Extend environment to include pi , 0q, p0, jq, i , j ě 0, W p0, 0q “ 0.
j
i1
1
X p1, 0qY p0, 1q
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 ` bjq
Passage time in this environment isdenoted Gzpi , jq.
X pi , jq “ Gzpi , jq ´ Gzpi ´ 1, jq
Y pi , jq “ Gzpi , jq ´ Gzpi , j ´ 1q
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Stationary model
Extend environment to include pi , 0q, p0, jq, i , j ě 0, W p0, 0q “ 0.
j
i1
1
X p1, 0qY p0, 1q
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 ` bjq
Passage time in this environment isdenoted Gzpi , jq.
X pi , jq “ Gzpi , jq ´ Gzpi ´ 1, jq
Y pi , jq “ Gzpi , jq ´ Gzpi , j ´ 1q
Chris Janjigian Large deviations for certain inhomogeneous corner growth models 8/29
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Stationary model
Extend environment to include pi , 0q, p0, jq, i , j ě 0, W p0, 0q “ 0.
� � " '
j
i1
1
X p1, 0qY p0, 1q
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 ` bjq
Passage time in this environment isdenoted Gzpi , jq.
X pi , jq “ Gzpi , jq ´ Gzpi ´ 1, jq
Y pi , jq “ Gzpi , jq ´ Gzpi , j ´ 1q
Chris Janjigian Large deviations for certain inhomogeneous corner growth models 8/29
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Stationary model
Extend environment to include pi , 0q, p0, jq, i , j ě 0, W p0, 0q “ 0.
�
"
.
:
F
j
i1
1
X p1, 0qY p0, 1q
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 ` bjq
Passage time in this environment isdenoted Gzpi , jq.
X pi , jq “ Gzpi , jq ´ Gzpi ´ 1, jq
Y pi , jq “ Gzpi , jq ´ Gzpi , j ´ 1q
Chris Janjigian Large deviations for certain inhomogeneous corner growth models 8/29
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Stationary model
Extend environment to include pi , 0q, p0, jq, i , j ě 0, W p0, 0q “ 0.
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
j
i1
1
X p1, 0qY p0, 1q
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 ` bjq
Passage time in this environment isdenoted Gzpi , jq.
X pi , jq “ Gzpi , jq ´ Gzpi ´ 1, jq
Y pi , jq “ Gzpi , jq ´ Gzpi , j ´ 1q
Chris Janjigian Large deviations for certain inhomogeneous corner growth models 8/29
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Stationary model
Extend environment to include pi , 0q, p0, jq, i , j ě 0, W p0, 0q “ 0.
j
i1
1
X p1, 0qY p0, 1q
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 ` bjq
Passage time in this environment isdenoted Gzpi , jq.
X pi , jq “ Gzpi , jq ´ Gzpi ´ 1, jq
Y pi , jq “ Gzpi , jq ´ Gzpi , j ´ 1q
Chris Janjigian Large deviations for certain inhomogeneous corner growth models 8/29
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Stationary model
Extend environment to include pi , 0q, p0, jq, i , j ě 0, W p0, 0q “ 0.
j
i1
1
X p1, 0qY p0, 1q
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 ` bjq
Passage time in this environment isdenoted Gzpi , jq.
X pi , jq “ Gzpi , jq ´ Gzpi ´ 1, jq
Y pi , jq “ Gzpi , jq ´ Gzpi , j ´ 1q
Chris Janjigian Large deviations for certain inhomogeneous corner growth models 8/29
![Page 18: Large deviations for certain inhomogeneous corner growth ... · Outline 1 The (totally asymmetric) corner growth model 1 Questions: general homogeneous shape theorem and large deviations](https://reader034.fdocuments.net/reader034/viewer/2022050608/5faf1f72df34d800c10249aa/html5/thumbnails/18.jpg)
Stationary model
Extend environment to include pi , 0q, p0, jq, i , j ě 0, W p0, 0q “ 0.
j
i1
1
X p1, 0qY p0, 1q
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 ` bjq
Passage time in this environment isdenoted Gzpi , jq.
X pi , jq “ Gzpi , jq ´ Gzpi ´ 1, jq
Y pi , jq “ Gzpi , jq ´ Gzpi , j ´ 1q
Chris Janjigian Large deviations for certain inhomogeneous corner growth models 8/29
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Stationary model - exponential lemma
Lemma
Suppose that pX ,Y ,W q are mutually independent exponential randomvariables with means pa` zq´1, pb ´ zq´1, pa` bq´1 and define
X “ pX ´ Y q` `W , Y “ pY ´ X q` `W , W “ X ^ Y .
Then
pX , Y , W qd“ pX ,Y ,W q.
Proof: Compute the Laplace transform of pX , Y , W q.
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Stationary model - corner flipping
W pi , jq „ Exppai ` bjq
Y pi ´ 1, jq „ Exppbj ´ zq
X pi , j ´ 1q „ Exppai ` zq
X pi , jq „ Exppai ` zq
Y pi , jq „ Exppbj ´ zq
|W pi ´ 1, j ´ 1q „ Exppai ` bjq
X pi , jq “ pX pi , j ´ 1q ´ Y pi ´ 1, jqq` `W pi , jq
Y pi , jq “ pY pi ´ 1, jq ´ X pi , j ´ 1qq` `W pi , jq
|W pi ´ 1, j ´ 1q “ X pi , j ´ 1q ^ Y pi ´ 1, jq
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Stationary model - corner flipping
W pi , jq „ Exppai ` bjq
Y pi ´ 1, jq „ Exppbj ´ zq
X pi , j ´ 1q „ Exppai ` zq
X pi , jq „ Exppai ` zq
Y pi , jq „ Exppbj ´ zq
|W pi ´ 1, j ´ 1q „ Exppai ` bjq
X pi , jq “ pX pi , j ´ 1q ´ Y pi ´ 1, jqq` `W pi , jq
Y pi , jq “ pY pi ´ 1, jq ´ X pi , j ´ 1qq` `W pi , jq
|W pi ´ 1, j ´ 1q “ X pi , j ´ 1q ^ Y pi ´ 1, jq
Chris Janjigian Large deviations for certain inhomogeneous corner growth models 10/29
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Stationary model - corner flipping
W pi , jq „ Exppai ` bjq
Y pi ´ 1, jq „ Exppbj ´ zq
X pi , j ´ 1q „ Exppai ` zq
X pi , jq „ Exppai ` zq
Y pi , jq „ Exppbj ´ zq
|W pi ´ 1, j ´ 1q „ Exppai ` bjq
X pi , jq “ pX pi , j ´ 1q ´ Y pi ´ 1, jqq` `W pi , jq
Y pi , jq “ pY pi ´ 1, jq ´ X pi , j ´ 1qq` `W pi , jq
|W pi ´ 1, j ´ 1q “ X pi , j ´ 1q ^ Y pi ´ 1, jq
Chris Janjigian Large deviations for certain inhomogeneous corner growth models 10/29
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Stationary model - down right paths
j
i
X pi , 0q “W pi , 0q „ Exppai ` zq
Y p0, jq “W p0, jq „ Exppbj ´ zq
By induction, mutualindependence and distributionsare preserved along down-rightedge paths.
X pi , jq „ Exppai ` zq
Y pi , jq „ Exppbj ´ zq.
Chris Janjigian Large deviations for certain inhomogeneous corner growth models 11/29
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Stationary model - down right paths
j
i
X pi , 0q “W pi , 0q „ Exppai ` zq
Y p0, jq “W p0, jq „ Exppbj ´ zq
By induction, mutualindependence and distributionsare preserved along down-rightedge paths.
X pi , jq „ Exppai ` zq
Y pi , jq „ Exppbj ´ zq.
Chris Janjigian Large deviations for certain inhomogeneous corner growth models 11/29
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Stationary model - down right paths
j
i
X pi , 0q “W pi , 0q „ Exppai ` zq
Y p0, jq “W p0, jq „ Exppbj ´ zq
By induction, mutualindependence and distributionsare preserved along down-rightedge paths.
X pi , jq „ Exppai ` zq
Y pi , jq „ Exppbj ´ zq.
Chris Janjigian Large deviations for certain inhomogeneous corner growth models 11/29
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Stationary model - down right paths
j
i
X pi , 0q “W pi , 0q „ Exppai ` zq
Y p0, jq “W p0, jq „ Exppbj ´ zq
By induction, mutualindependence and distributionsare preserved along down-rightedge paths.
X pi , jq „ Exppai ` zq
Y pi , jq „ Exppbj ´ zq.
Chris Janjigian Large deviations for certain inhomogeneous corner growth models 11/29
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Stationary model - down right paths
j
i
X pi , 0q “W pi , 0q „ Exppai ` zq
Y p0, jq “W p0, jq „ Exppbj ´ zq
By induction, mutualindependence and distributionsare preserved along down-rightedge paths.
X pi , jq „ Exppai ` zq
Y pi , jq „ Exppbj ´ zq.
Chris Janjigian Large deviations for certain inhomogeneous corner growth models 11/29
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Random environment
W pi , jq ind.„ Exppai ` bjq
�
a1 . . . ai . . .
b1
...
bj
...
a “ pai q, b “ pbjq indep. i.i.d.,Era1 ` b1s ă 8, ai , bj ě c ą 0 (canbe weaker)
Pa,b: conditioned on pa,bq,
W pi , jqind.„ Exppai ` bjq.
P: average Pa,b over pa,bq:Pp¨q “ ErPa,bp¨qs.
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Random environment
W pi , jq ind.„ Exppai ` bjq
�
a1 . . . ai . . .
b1
...
bj
...
a “ pai q, b “ pbjq indep. i.i.d.,Era1 ` b1s ă 8, ai , bj ě c ą 0 (canbe weaker)
Pa,b: conditioned on pa,bq,
W pi , jqind.„ Exppai ` bjq.
P: average Pa,b over pa,bq:Pp¨q “ ErPa,bp¨qs.
Chris Janjigian Large deviations for certain inhomogeneous corner growth models 12/29
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Random environment
W pi , jq ind.„ Exppai ` bjq
�
a1 . . . ai . . .
b1
...
bj
...
a “ pai q, b “ pbjq indep. i.i.d.,Era1 ` b1s ă 8, ai , bj ě c ą 0 (canbe weaker)
Pa,b: conditioned on pa,bq,
W pi , jqind.„ Exppai ` bjq.
P: average Pa,b over pa,bq:Pp¨q “ ErPa,bp¨qs.
Chris Janjigian Large deviations for certain inhomogeneous corner growth models 12/29
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Random environment
W pi , jq ind.„ Exppai ` bjq
�
a1 . . . ai . . .
b1
...
bj
...
Key points:
Pa,b: indep., not ident. dist.:if i ‰ i 1 or j ‰ j 1
W pi , jqd‰W pi 1, j 1q (usually)
W pi , jq KW pi 1, j 1q
P: ident. dist., not indep.:
W pi , jqd“W pi 1, j 1q.
If i “ i 1 or j “ j 1
CovpW pi , jq,W pi 1, j 1qq ‰ 0.
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Random environment
W pi , jq ind.„ Exppai ` bjq
�
a1 . . . ai . . .
b1
...
bj
...
Key points:
Pa,b: indep., not ident. dist.:if i ‰ i 1 or j ‰ j 1
W pi , jqd‰W pi 1, j 1q (usually)
W pi , jq KW pi 1, j 1q
P: ident. dist., not indep.:
W pi , jqd“W pi 1, j 1q.
If i “ i 1 or j “ j 1
CovpW pi , jq,W pi 1, j 1qq ‰ 0.
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Random environment
W pi , jq ind.„ Exppai ` bjq
�
a1 . . . ai . . .
b1
...
bj
...
Key points:
Pa,b: indep., not ident. dist.:if i ‰ i 1 or j ‰ j 1
W pi , jqd‰W pi 1, j 1q (usually)
W pi , jq KW pi 1, j 1q
P: ident. dist., not indep.:
W pi , jqd“W pi 1, j 1q.
If i “ i 1 or j “ j 1
CovpW pi , jq,W pi 1, j 1qq ‰ 0.
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Random environment
W pi , jq ind.„ Exppai ` bjq
�
a1 . . . ai . . .
b1
...
bj
...
Key points:
Pa,b: indep., not ident. dist.:if i ‰ i 1 or j ‰ j 1
W pi , jqd‰W pi 1, j 1q (usually)
W pi , jq KW pi 1, j 1q
P: ident. dist., not indep.:
W pi , jqd“W pi 1, j 1q.
If i “ i 1 or j “ j 1
CovpW pi , jq,W pi 1, j 1qq ‰ 0.
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Random environment
W pi , jq ind.„ Exppai ` bjq
�
a1 . . . ai . . .
b1
...
bj
...
Key points:
Pa,b: indep., not ident. dist.:if i ‰ i 1 or j ‰ j 1
W pi , jqd‰W pi 1, j 1q (usually)
W pi , jq KW pi 1, j 1q
P: ident. dist., not indep.:
W pi , jqd“W pi 1, j 1q.
If i “ i 1 or j “ j 1
CovpW pi , jq,W pi 1, j 1qq ‰ 0.
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Stationary shape function
Lemma
For s, t ą 0, Pz almost surely and for almost all pa,bq Pza,b almost surely
gzps, tq :“ limnÑ8
1
nGzpt ns u, t nt uq “ s E
„
1
a1 ` z
` t E
„
1
b1 ´ z
.
Proof:
pGzpn, nq “ÿ
1ďiďn
”
pGzpi , 0q ´ pGzpi ´ 1, 0qı
`ÿ
1ďjďn
”
pGzpn, jq ´ pGzpn, j ´ 1qı
“ÿ
1ďiďn
X pi , 0q `ÿ
1ďjďn
Y pn, jq
.
These sums are marginally i.i.d. under Pz (not mutually indep. underPz
a,b or Pz).
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Stationary shape function
Lemma
For s, t ą 0, Pz almost surely and for almost all pa,bq Pza,b almost surely
gzps, tq :“ limnÑ8
1
nGzpt ns u, t nt uq “ s E
„
1
a1 ` z
` t E
„
1
b1 ´ z
.
Proof:
pGzpn, nq “ÿ
1ďiďn
”
pGzpi , 0q ´ pGzpi ´ 1, 0qı
`ÿ
1ďjďn
”
pGzpn, jq ´ pGzpn, j ´ 1qı
“ÿ
1ďiďn
X pi , 0q `ÿ
1ďjďn
Y pn, jq.
These sums are marginally i.i.d. under Pz (not mutually indep. underPz
a,b or Pz).
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Stationary shape function
Lemma
For s, t ą 0, Pz almost surely and for almost all pa,bq Pza,b almost surely
gzps, tq :“ limnÑ8
1
nGzpt ns u, t nt uq “ s E
„
1
a1 ` z
` t E
„
1
b1 ´ z
.
Proof:
pGzpn, nq “ÿ
1ďiďn
”
pGzpi , 0q ´ pGzpi ´ 1, 0qı
`ÿ
1ďjďn
”
pGzpn, jq ´ pGzpn, j ´ 1qı
“ÿ
1ďiďn
X pi , 0q `ÿ
1ďjďn
Y pn, jq.
These sums are marginally i.i.d. under Pz (not mutually indep. underPz
a,b or Pz).
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Stationary shape function
Lemma
For s, t ą 0, Pz almost surely and for almost all pa,bq Pza,b almost surely
gzps, tq :“ limnÑ8
1
nGzpt ns u, t nt uq “ s E
„
1
a1 ` z
` t E
„
1
b1 ´ z
.
Proof:
pGzpn, nq “ÿ
1ďiďn
”
pGzpi , 0q ´ pGzpi ´ 1, 0qı
`ÿ
1ďjďn
”
pGzpn, jq ´ pGzpn, j ´ 1qı
“ÿ
1ďiďn
X pi , 0q `ÿ
1ďjďn
Y pn, jq.
These sums are marginally i.i.d. under Pz (not mutually indep. underPz
a,b or Pz).
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Shape function
Notation:¯α “ essinfta1u,
¯β “ essinftb1u.
Theorem (Emrah ’15)
For s, t ą 0, P almost surely and for almost all pa,bq Pa,b almost surely
gps, tq :“ limnÑ8
1
nG pt ns u, t nt uq “ min
´¯αďzď
¯β
"
s E
„
1
a1 ` z
` t E
„
1
b1 ´ z
*
Remark:
These results also hold if a and b are both separately ergodic, ratherthan a pair of independent i.i.d. sequences. The formulas onlydepend on marginal distributions of a1 and b1 separately.
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Shape function
Notation:¯α “ essinfta1u,
¯β “ essinftb1u.
Theorem (Emrah ’15)
For s, t ą 0, P almost surely and for almost all pa,bq Pa,b almost surely
gps, tq :“ limnÑ8
1
nG pt ns u, t nt uq “ min
´¯αďzď
¯β
"
s E
„
1
a1 ` z
` t E
„
1
b1 ´ z
*
Remark:
These results also hold if a and b are both separately ergodic, ratherthan a pair of independent i.i.d. sequences. The formulas onlydepend on marginal distributions of a1 and b1 separately.
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Shape function
Notation:¯α “ essinfta1u,
¯β “ essinftb1u.
Theorem (Emrah ’15)
For s, t ą 0, P almost surely and for almost all pa,bq Pa,b almost surely
gps, tq :“ limnÑ8
1
nG pt ns u, t nt uq “ min
´¯αďzď
¯β
"
s E
„
1
a1 ` z
` t E
„
1
b1 ´ z
*
Remark:
These results also hold if a and b are both separately ergodic, ratherthan a pair of independent i.i.d. sequences. The formulas onlydepend on marginal distributions of a1 and b1 separately.
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Asymptotic shape of the cluster
t
s0
g ď 1
S
S1
S2
Key properties of the shape functiongps, tq:
g is strictly concave in S , linearin S1 and S2.
S1,S2 ‰ H iff
E rpa1 ´¯αq´2s ă 8 pS1q
E rpb1 ´¯βq´2s ă 8 pS2q
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Asymptotic shape of the cluster
t
s0
g ď 1
S
S1
S2
Key properties of the shape functiongps, tq:
g is strictly concave in S , linearin S1 and S2.
S1,S2 ‰ H iff
E rpa1 ´¯αq´2s ă 8 pS1q
E rpb1 ´¯βq´2s ă 8 pS2q
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Asymptotic shape of the cluster
t
s0
g ď 1
S
S1
S2
Key properties of the shape functiongps, tq:
g is strictly concave in S , linearin S1 and S2.
S1,S2 ‰ H iff
E rpa1 ´¯αq´2s ă 8 pS1q
E rpb1 ´¯βq´2s ă 8 pS2q
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Asymptotic shape of the cluster
t
s0
g ď 1
S
S1
S2
Key properties of the shape functiongps, tq:
g is strictly concave in S , linearin S1 and S2.
S1,S2 ‰ H iff
E rpa1 ´¯αq´2s ă 8 pS1q
E rpb1 ´¯βq´2s ă 8 pS2q
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Quenched right tail rate function
Theorem
For almost all pa,bq, for any s, t ą 0 and r ě gps, tq
Js,tprq “ limnÑ8
´n´1 logPa,b
`
n´1G pt ns u, t nt uq ě r˘
“ supλPp0,
¯α`
¯βq
zPp´¯α,
¯β´λq
"
rλ` s E log
ˆ
a1 ` z
a1 ` z ` λ
˙
´ t E log
ˆ
b1 ´ z
b1 ´ z ´ λ
˙*
Remarks:
The same theorem also holds if pa,bq is totally ergodic rather thana pair of independent i.i.d. sequences. Again, these formulas onlydepend on marginal distributions of a1 and b1 separately.
Rate n LDP for n´1G pt ns u, t nt uq under Pa,b with rate function
Is,tprq “ Js,tprq1trěgps,tqu `81trăgps,tqu.
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Quenched right tail rate function
Theorem
For almost all pa,bq, for any s, t ą 0 and r ě gps, tq
Js,tprq “ limnÑ8
´n´1 logPa,b
`
n´1G pt ns u, t nt uq ě r˘
“ supλPp0,
¯α`
¯βq
zPp´¯α,
¯β´λq
"
rλ` s E log
ˆ
a1 ` z
a1 ` z ` λ
˙
´ t E log
ˆ
b1 ´ z
b1 ´ z ´ λ
˙*
Remarks:
The same theorem also holds if pa,bq is totally ergodic rather thana pair of independent i.i.d. sequences. Again, these formulas onlydepend on marginal distributions of a1 and b1 separately.
Rate n LDP for n´1G pt ns u, t nt uq under Pa,b with rate function
Is,tprq “ Js,tprq1trěgps,tqu `81trăgps,tqu.
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Quenched right tail rate function
Theorem
For almost all pa,bq, for any s, t ą 0 and r ě gps, tq
Js,tprq “ limnÑ8
´n´1 logPa,b
`
n´1G pt ns u, t nt uq ě r˘
“ supλPp0,
¯α`
¯βq
zPp´¯α,
¯β´λq
"
rλ` s E log
ˆ
a1 ` z
a1 ` z ` λ
˙
´ t E log
ˆ
b1 ´ z
b1 ´ z ´ λ
˙*
Remarks:
The same theorem also holds if pa,bq is totally ergodic rather thana pair of independent i.i.d. sequences. Again, these formulas onlydepend on marginal distributions of a1 and b1 separately.
Rate n LDP for n´1G pt ns u, t nt uq under Pa,b with rate function
Is,tprq “ Js,tprq1trěgps,tqu `81trăgps,tqu.
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Expected fluctuations
t
s0
l1
l2S
S1
S2
Quenched fluct. are TWGUE inS , but not in S1,S2.
Q1: Can we “see” differentscaling exponents in the ratefunctions?
Q2: What happens whenps, tq P l1, l2?
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Expected fluctuations
t
s0
l1
l2S
S1
S2
Quenched fluct. are TWGUE inS , but not in S1,S2.
Q1: Can we “see” differentscaling exponents in the ratefunctions?
Q2: What happens whenps, tq P l1, l2?
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Expected fluctuations
t
s0
l1
l2S
S1
S2
Quenched fluct. are TWGUE inS , but not in S1,S2.
Q1: Can we “see” differentscaling exponents in the ratefunctions?
Q2: What happens whenps, tq P l1, l2?
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Scaling and the quenched rate functions
Proposition
For any s, t ą 0, let ζ P r´¯α,
¯βs solve (uniquely) gζps, tq “ gps, tq. As
ε Ó 0, there are explicit values C1,C2,C3 depending on s, t, ζ such that
Js,tpgps, tq ` εq “
$
’
’
’
’
’
’
’
’
’
&
’
’
’
’
’
’
’
’
’
%
C1ps, tq ε2 ` opε2q ps, tq P S1
12C2ps, t,
¯αq ε3{2 ` opε3{2q ps, tq P l1
C2ps, t, ζq ε3{2 ` opε3{2q ps, tq P S
12C2ps, t,
¯βq ε3{2 ` opε3{2q ps, tq P l2
C3ps, tq ε2 ` opε2q ps, tq P S2
.
t
s
l1
l2S
S1
S2
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Scaling and the quenched rate functions
Proposition
For any s, t ą 0, let ζ P r´¯α,
¯βs solve (uniquely) gζps, tq “ gps, tq. As
ε Ó 0, there are explicit values C1,C2,C3 depending on s, t, ζ such that
Js,tpgps, tq ` εq “
$
’
’
’
’
’
’
’
’
’
&
’
’
’
’
’
’
’
’
’
%
C1ps, tq ε2 ` opε2q ps, tq P S1
12C2ps, t,
¯αq ε3{2 ` opε3{2q ps, tq P l1
C2ps, t, ζq ε3{2 ` opε3{2q ps, tq P S
12C2ps, t,
¯βq ε3{2 ` opε3{2q ps, tq P l2
C3ps, tq ε2 ` opε2q ps, tq P S2
.
t
s
l1
l2S
S1
S2
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Scaling and the quenched rate functions
Heuristically consistent with TWGUE fluct in S .
Take ps, tq P S and set
C “ s E
„
1
pa` ζq3
` t E
„
1
pb ´ ζq3
“1
2B2zgzps, tq
ˇ
ˇ
z“ζ
In the notation of the previous result, C2ps, t, ζq “ 4{3C´1{2. For n
large and large enough r (but not O(n23 )), we might expect
Pa,bpG pt ns u, t nt uq ´ ngps, tq ě n13 C
13 rq « e´n Js,tpgps,tq`C
13 n´
23 rq
« e´43 C´ 1
2 pC13 n´
23 rq
32 n “ e´
43 r
32
which agrees with the leading order TWGUE right tail.
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Scaling and the quenched rate functions
Heuristically consistent with TWGUE fluct in S . Take ps, tq P S and set
C “ s E
„
1
pa` ζq3
` t E
„
1
pb ´ ζq3
“1
2B2zgzps, tq
ˇ
ˇ
z“ζ
In the notation of the previous result, C2ps, t, ζq “ 4{3C´1{2.
For n
large and large enough r (but not O(n23 )), we might expect
Pa,bpG pt ns u, t nt uq ´ ngps, tq ě n13 C
13 rq « e´n Js,tpgps,tq`C
13 n´
23 rq
« e´43 C´ 1
2 pC13 n´
23 rq
32 n “ e´
43 r
32
which agrees with the leading order TWGUE right tail.
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Scaling and the quenched rate functions
Heuristically consistent with TWGUE fluct in S . Take ps, tq P S and set
C “ s E
„
1
pa` ζq3
` t E
„
1
pb ´ ζq3
“1
2B2zgzps, tq
ˇ
ˇ
z“ζ
In the notation of the previous result, C2ps, t, ζq “ 4{3C´1{2. For n
large and large enough r (but not O(n23 )), we might expect
Pa,bpG pt ns u, t nt uq ´ ngps, tq ě n13 C
13 rq
« e´n Js,tpgps,tq`C13 n´
23 rq
« e´43 C´ 1
2 pC13 n´
23 rq
32 n “ e´
43 r
32
which agrees with the leading order TWGUE right tail.
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Scaling and the quenched rate functions
Heuristically consistent with TWGUE fluct in S . Take ps, tq P S and set
C “ s E
„
1
pa` ζq3
` t E
„
1
pb ´ ζq3
“1
2B2zgzps, tq
ˇ
ˇ
z“ζ
In the notation of the previous result, C2ps, t, ζq “ 4{3C´1{2. For n
large and large enough r (but not O(n23 )), we might expect
Pa,bpG pt ns u, t nt uq ´ ngps, tq ě n13 C
13 rq « e´n Js,tpgps,tq`C
13 n´
23 rq
« e´43 C´ 1
2 pC13 n´
23 rq
32 n “ e´
43 r
32
which agrees with the leading order TWGUE right tail.
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Scaling and the quenched rate functions
Heuristically consistent with TWGUE fluct in S . Take ps, tq P S and set
C “ s E
„
1
pa` ζq3
` t E
„
1
pb ´ ζq3
“1
2B2zgzps, tq
ˇ
ˇ
z“ζ
In the notation of the previous result, C2ps, t, ζq “ 4{3C´1{2. For n
large and large enough r (but not O(n23 )), we might expect
Pa,bpG pt ns u, t nt uq ´ ngps, tq ě n13 C
13 rq « e´n Js,tpgps,tq`C
13 n´
23 rq
« e´43 C´ 1
2 pC13 n´
23 rq
32 n “ e´
43 r
32
which agrees with the leading order TWGUE right tail.
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Annealed large deviations
Theorem
For s, t ą 0 and r ě gps, tq,
Js,tprq “ limnÑ8
´n´1 logP`
n´1G pt ns u, t nt uq ě r˘
“ supλPp0,
¯α`
¯βq
zPp´¯α,
¯β´λq
"
rλ´ s log E
„
a1 ` z ` λ
a1 ` z
´ t log E
„
b1 ´ z
b1 ´ z ´ λ
*
Remark:
We do not have the rate n left tail rate function in this case, but wecan show existence of r P p0, gps, tqq with
lim sup´1
nP`
n´1G pt ns u, t nt uq ď r˘
ă 8.
(i.e. there are rate n annealed left tail large deviations)
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Annealed large deviations
Theorem
For s, t ą 0 and r ě gps, tq,
Js,tprq “ limnÑ8
´n´1 logP`
n´1G pt ns u, t nt uq ě r˘
“ supλPp0,
¯α`
¯βq
zPp´¯α,
¯β´λq
"
rλ´ s log E
„
a1 ` z ` λ
a1 ` z
´ t log E
„
b1 ´ z
b1 ´ z ´ λ
*
Remark:
We do not have the rate n left tail rate function in this case, but wecan show existence of r P p0, gps, tqq with
lim sup´1
nP`
n´1G pt ns u, t nt uq ď r˘
ă 8.
(i.e. there are rate n annealed left tail large deviations)
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Scaling and the annealed rate functions
Proposition
For any s, t ą 0, let ζ P r´¯α,
¯βs solve (uniquely) gζps, tq “ gps, tq. As
ε Ó 0, there are explicit values C1,C2,C3 depending on s, t, ζ such that
Js,tpgps, tq ` εq “
$
’
’
&
’
’
%
C1ps, tq ε2 ` opε2q ps, tq P S1
C2ps, t, ζq ε2 ` opε2q ps, tq P S , l1, l2
C3ps, tq ε2 ` opε2q ps, tq P S2
.
t
s
l1
l2S
S1
S2
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Scaling and the annealed rate functions
Proposition
For any s, t ą 0, let ζ P r´¯α,
¯βs solve (uniquely) gζps, tq “ gps, tq. As
ε Ó 0, there are explicit values C1,C2,C3 depending on s, t, ζ such that
Js,tpgps, tq ` εq “
$
’
’
&
’
’
%
C1ps, tq ε2 ` opε2q ps, tq P S1
C2ps, t, ζq ε2 ` opε2q ps, tq P S , l1, l2
C3ps, tq ε2 ` opε2q ps, tq P S2
.
t
s
l1
l2S
S1
S2
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Variational connection for right tail
Theorem
For any s, t ą 0 and r ě gps, tq,
Js,tprq “ infν1,ν2
Iν1,ν2s,t prq ` s Hpν1|αq ` t Hpν2|βq
(
.
A unique minimizing pair pν1, ν2q exists. The equality
Js,tprq “ Iν1,ν2s,t prq ` s Hpν1|αq ` t Hpν2|βq
holds if and only if
dν1
dαpaq 9
a` z‹`λ‹a` z‹
,dν2
dβpbq 9
b ´ z‹b ´ z‹´λ‹
where z‹ and λ‹ are the unique z‹, λ‹ withλ‹ P r0,
¯α`
¯βs, z‹ P r´
¯α,
¯β ´ λ‹s satisfying
Js,tprq “ r λ‹´s log Eα„
a` z‹`λ‹a` z‹
´ t log Eβ„
b ´ z‹b ´ z‹´λ‹
.
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Variational connection for right tail
Theorem
For any s, t ą 0 and r ě gps, tq,
Js,tprq “ infν1,ν2
Iν1,ν2s,t prq ` s Hpν1|αq ` t Hpν2|βq
(
.
A unique minimizing pair pν1, ν2q exists.
The equality
Js,tprq “ Iν1,ν2s,t prq ` s Hpν1|αq ` t Hpν2|βq
holds if and only if
dν1
dαpaq 9
a` z‹`λ‹a` z‹
,dν2
dβpbq 9
b ´ z‹b ´ z‹´λ‹
where z‹ and λ‹ are the unique z‹, λ‹ withλ‹ P r0,
¯α`
¯βs, z‹ P r´
¯α,
¯β ´ λ‹s satisfying
Js,tprq “ r λ‹´s log Eα„
a` z‹`λ‹a` z‹
´ t log Eβ„
b ´ z‹b ´ z‹´λ‹
.
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Variational connection for right tail
Theorem
For any s, t ą 0 and r ě gps, tq,
Js,tprq “ infν1,ν2
Iν1,ν2s,t prq ` s Hpν1|αq ` t Hpν2|βq
(
.
A unique minimizing pair pν1, ν2q exists. The equality
Js,tprq “ Iν1,ν2s,t prq ` s Hpν1|αq ` t Hpν2|βq
holds if and only if
dν1
dαpaq 9
a` z‹`λ‹a` z‹
,dν2
dβpbq 9
b ´ z‹b ´ z‹´λ‹
where z‹ and λ‹ are the unique z‹, λ‹ withλ‹ P r0,
¯α`
¯βs, z‹ P r´
¯α,
¯β ´ λ‹s satisfying
Js,tprq “ r λ‹´s log Eα„
a` z‹`λ‹a` z‹
´ t log Eβ„
b ´ z‹b ´ z‹´λ‹
.
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Variational connection for right tail
Theorem
For any s, t ą 0 and r ě gps, tq,
Js,tprq “ infν1,ν2
Iν1,ν2s,t prq ` s Hpν1|αq ` t Hpν2|βq
(
.
A unique minimizing pair pν1, ν2q exists. The equality
Js,tprq “ Iν1,ν2s,t prq ` s Hpν1|αq ` t Hpν2|βq
holds if and only if
dν1
dαpaq 9
a` z‹`λ‹a` z‹
,dν2
dβpbq 9
b ´ z‹b ´ z‹´λ‹
where z‹ and λ‹ are the unique z‹, λ‹ withλ‹ P r0,
¯α`
¯βs, z‹ P r´
¯α,
¯β ´ λ‹s satisfying
Js,tprq “ r λ‹´s log Eα„
a` z‹`λ‹a` z‹
´ t log Eβ„
b ´ z‹b ´ z‹´λ‹
.
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Proof sketch: finite n variational problem
Denote by θi,j the shift on the environment by pi , jq. We have
pGzpn, nq “ maxt max1ďkďn
tG pn ´ k ` 1, nq ˝ θk´1,0 ` pGzpk, 0q,
max1ď`ďn
tG pn, n ´ `` 1q ˝ θ0,`´1 ` pGzp0, `quu.
k “ 2
` “ 2
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Proof sketch: finite n variational problem
Denote by θi,j the shift on the environment by pi , jq. We have
pGzpn, nq “ maxt max1ďkďn
tG pn ´ k ` 1, nq ˝ θk´1,0 ` pGzpk, 0q,
max1ď`ďn
tG pn, n ´ `` 1q ˝ θ0,`´1 ` pGzp0, `quu.
k “ 2
` “ 2
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Proof sketch: finite n variational problem
Denote by θi,j the shift on the environment by pi , jq. We have
pGzpn, nq “ maxt max1ďkďn
tG pn ´ k ` 1, nq ˝ θk´1,0 ` pGzpk, 0q,
max1ď`ďn
tG pn, n ´ `` 1q ˝ θ0,`´1 ` pGzp0, `quu.
k “ 2
` “ 2
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Proof sketch: finite n variational problem
Denote by θi,j the shift on the environment by pi , jq. We have
pGzpn, nq “ maxt max1ďkďn
tG pn ´ k ` 1, nq ˝ θk´1,0 ` pGzpk, 0q,
max1ď`ďn
tG pn, n ´ `` 1q ˝ θ0,`´1 ` pGzp0, `quu.
k “ 2
` “ 2
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Proof sketch: finite n variational problem
Denote by θi,j the shift on the environment by pi , jq. We have
pGzpn, nq “ maxt max1ďkďn
tG pn ´ k ` 1, nq ˝ θk´1,0 ` pGzpk, 0q,
max1ď`ďn
tG pn, n ´ `` 1q ˝ θ0,`´1 ` pGzp0, `quu.
k “ 2
` “ 2
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Proof sketch: finite n variational problem
Denote by θi,j the shift on the environment by pi , jq. We have
pGzpn, nq “ maxt max1ďkďn
tG pn ´ k ` 1, nq ˝ θk´1,0 ` pGzpk, 0q,
max1ď`ďn
tG pn, n ´ `` 1q ˝ θ0,`´1 ` pGzp0, `quu.
k “ 2
` “ 2
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Proof sketch: finite n variational problem
Denote by θi,j the shift on the environment by pi , jq. We have
pGzpn, nq “ maxt max1ďkďn
tG pn ´ k ` 1, nq ˝ θk´1,0 ` pGzpk, 0q,
max1ď`ďn
tG pn, n ´ `` 1q ˝ θ0,`´1 ` pGzp0, `quu.
k “ 2
` “ 2
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Proof sketch: finite n variational problem
Denote by θi,j the shift on the environment by pi , jq. We have
pGzpn, nq “ maxt max1ďkďn
tG pn ´ k ` 1, nq ˝ θk´1,0 ` pGzpk, 0q,
max1ď`ďn
tG pn, n ´ `` 1q ˝ θ0,`´1 ` pGzp0, `quu.
k “ 2
` “ 2
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Proof sketch: finite n variational problem
Denote by θi,j the shift on the environment by pi , jq. We have
pGzpn, nq “ maxt max1ďkďn
tG pn ´ k ` 1, nq ˝ θk´1,0 ` pGzpk, 0q,
max1ď`ďn
tG pn, n ´ `` 1q ˝ θ0,`´1 ` pGzp0, `quu.
k “ 2
` “ 2
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Proof sketch: telescoping sum
pGzpn, nq “ÿ
1ďiďn
pGzpi , 0q ´ pGzpi ´ 1, 0q `ÿ
1ďjďn
pGzpn, jq ´ pGzpn, j ´ 1q
“ÿ
1ďiďn
X pi , 0q `ÿ
1ďjďn
Y pn, jq (sums not indep.)
.
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Proof sketch: telescoping sum
pGzpn, nq “ÿ
1ďiďn
pGzpi , 0q ´ pGzpi ´ 1, 0q `ÿ
1ďjďn
pGzpn, jq ´ pGzpn, j ´ 1q
“ÿ
1ďiďn
X pi , 0q `ÿ
1ďjďn
Y pn, jq (sums not indep.).
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Proof sketch: variational problem for Js,t
Equate two expressions for Gzpn, nq and subtract theřn
i“1 X pi , 0q terms:
ÿ
1ďjďn
Y pn, jq “ max1ďkďn
tmaxtG pn ´ k ` 1, nq ˝ θk´1,0 ´ÿ
kăiďn
X pi , 0q,
G pn, n ´ k ` 1q ˝ θ0,k´1 ´ÿ
1ďiďn
X pi , 0q `ÿ
1ďjďk
Y p0, jquu.
For each z P p´¯α,
¯βq of the summands within the maxima are Pz
a,b-indep.Fix λ ą 0 and z P p´
¯α,
¯β ´ λq. For each k, we have
Eza,b
”
eλř
Y pn,jqı
ě Eza,b
”
eλGpn´k`1,nq˝θk´1,0
ı
Eza,b
„
e´λ
ř
kăiďn
Xpi,0q
Eza,b
”
eλř
Y pn,jqı
ě Eza,b
”
eλGpn,n´k`1q˝θ0,k´1
ı
Eza,b
„
e´λ
ř
1ďiďn
Xpi,0q
ˆ Eza,b
„
eλ
ř
1ďjďk
Y p0,jq
so LHS ě max of RHS over k.
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Proof sketch: variational problem for Js,t
Equate two expressions for Gzpn, nq and subtract theřn
i“1 X pi , 0q terms:
ÿ
1ďjďn
Y pn, jq “ max1ďkďn
tmaxtG pn ´ k ` 1, nq ˝ θk´1,0 ´ÿ
kăiďn
X pi , 0q,
G pn, n ´ k ` 1q ˝ θ0,k´1 ´ÿ
1ďiďn
X pi , 0q `ÿ
1ďjďk
Y p0, jquu.
For each z P p´¯α,
¯βq of the summands within the maxima are Pz
a,b-indep.Fix λ ą 0 and z P p´
¯α,
¯β ´ λq. For each k, we have
Eza,b
”
eλř
Y pn,jqı
ě Eza,b
”
eλGpn´k`1,nq˝θk´1,0
ı
Eza,b
„
e´λ
ř
kăiďn
Xpi,0q
Eza,b
”
eλř
Y pn,jqı
ě Eza,b
”
eλGpn,n´k`1q˝θ0,k´1
ı
Eza,b
„
e´λ
ř
1ďiďn
Xpi,0q
ˆ Eza,b
„
eλ
ř
1ďjďk
Y p0,jq
so LHS ě max of RHS over k.
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Proof sketch: variational problem for Js,t
Equate two expressions for Gzpn, nq and subtract theřn
i“1 X pi , 0q terms:
ÿ
1ďjďn
Y pn, jq “ max1ďkďn
tmaxtG pn ´ k ` 1, nq ˝ θk´1,0 ´ÿ
kăiďn
X pi , 0q,
G pn, n ´ k ` 1q ˝ θ0,k´1 ´ÿ
1ďiďn
X pi , 0q `ÿ
1ďjďk
Y p0, jquu.
For each z P p´¯α,
¯βq of the summands within the maxima are Pz
a,b-indep.Fix λ ą 0 and z P p´
¯α,
¯β ´ λq. For each k , we have
Eza,b
”
eλř
Y pn,jqı
ě Eza,b
”
eλGpn´k`1,nq˝θk´1,0
ı
Eza,b
„
e´λ
ř
kăiďn
Xpi,0q
Eza,b
”
eλř
Y pn,jqı
ě Eza,b
”
eλGpn,n´k`1q˝θ0,k´1
ı
Eza,b
„
e´λ
ř
1ďiďn
Xpi,0q
ˆ Eza,b
„
eλ
ř
1ďjďk
Y p0,jq
so LHS ě max of RHS over k .
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Proof sketch: variational problem for Js,t
Equate two expressions for Gzpn, nq and subtract theřn
i“1 X pi , 0q terms:
ÿ
1ďjďn
Y pn, jq “ max1ďkďn
tmaxtG pn ´ k ` 1, nq ˝ θk´1,0 ´ÿ
kăiďn
X pi , 0q,
G pn, n ´ k ` 1q ˝ θ0,k´1 ´ÿ
1ďiďn
X pi , 0q `ÿ
1ďjďk
Y p0, jquu.
For each z P p´¯α,
¯βq of the summands within the maxima are Pz
a,b-indep.Fix λ ą 0 and z P p´
¯α,
¯β ´ λq. For each k , we have
Eza,b
”
eλř
Y pn,jqı
ě Eza,b
”
eλGpn´k`1,nq˝θk´1,0
ı
Eza,b
„
e´λ
ř
kăiďn
Xpi,0q
Eza,b
”
eλř
Y pn,jqı
ě Eza,b
”
eλGpn,n´k`1q˝θ0,k´1
ı
Eza,b
„
e´λ
ř
1ďiďn
Xpi,0q
ˆ Eza,b
„
eλ
ř
1ďjďk
Y p0,jq
so LHS ě max of RHS over k .
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Proof sketch: variational problem for Js,t
Equate two expressions for Gzpn, nq and subtract theřn
i“1 X pi , 0q terms:
ÿ
1ďjďn
Y pn, jq “ max1ďkďn
tmaxtG pn ´ k ` 1, nq ˝ θk´1,0 ´ÿ
kăiďn
X pi , 0q,
G pn, n ´ k ` 1q ˝ θ0,k´1 ´ÿ
1ďiďn
X pi , 0q `ÿ
1ďjďk
Y p0, jquu.
For each z P p´¯α,
¯βq of the summands within the maxima are Pz
a,b-indep.Fix λ ą 0 and z P p´
¯α,
¯β ´ λq. For each k , we have
Eza,b
”
eλř
Y pn,jqı
ě Eza,b
”
eλGpn´k`1,nq˝θk´1,0
ı
Eza,b
„
e´λ
ř
kăiďn
Xpi,0q
Eza,b
”
eλř
Y pn,jqı
ě Eza,b
”
eλGpn,n´k`1q˝θ0,k´1
ı
Eza,b
„
e´λ
ř
1ďiďn
Xpi,0q
ˆ Eza,b
„
eλ
ř
1ďjďk
Y p0,jq
so LHS ě max of RHS over k .
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Proof sketch: variational problem for Js,t
Define
Ls,tpλq “ limnÑ8
1
nlogEa,b
”
eλGpt ns u,t nt uqı
For λ P p0,¯β ´ zq, if we apply limnÑ8 n´1 logEz
a,breλ¨s, the previous
inequality and another coming from max ďř
gives
E
„
logb1 ´ z
b1 ´ z ´ λ
“ sup0ďtď1
"
max
"
L1,tpλq ´ t E
„
loga1 ` z ` λ
a1 ` z
*
,
L1,tpλq ´ E
„
loga1 ` z ` λ
a1 ` z
` p1´ tqE
„
logb1 ´ z
b1 ´ z ´ λ
*
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Proof sketch: variational problem for Js,t
This variational problem can be inverted to solve for Ls,tpλq “
$
’
’
’
&
’
’
’
%
λgps, tq λ ď 0
min´
¯αďzď
¯β´λ
"
s E
„
loga1 ` z ` λ
a1 ` z
` t E
„
logb1 ´ z
b1 ´ z ´ λ
*
λ P p0,¯α`
¯βs
8 λ ą¯α`
¯β
Cannot use Gartner-Ellis (steepness can fail), but can verify that theLegendre transform gives the rate function.
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Thanks!
Chris Janjigian Large deviations for certain inhomogeneous corner growth models 29/29