Harmonic Pinnacles in the Discrete Gaussian Model

Harmonic Pinnacles in the
Discrete Gaussian Model
JOINT WORK WITH E. LUBETZKY AND A. SLY
Inhomogeneous Random Systems 2015
Fabio Martinelli
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The Discrete Gaussian model
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Inhomogeneous Random Systems, 2015
Fabio Martinelli, Univ. Roma Tre
The Discrete Gaussian model
• Family of surfaces models in the ’50 • Dubbed Discrete Gaussian Model by [Chui-­‐Weeks ’76] • Dual of the Villain XY model [Villain ’75] • Related by duality to the Coulomb gas model • its R-­‐valued analogue: β scales out Discrete Gaussian Free Field
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Fabio Martinelli, Univ. Roma Tre
DG surface: basic questions
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Fabio Martinelli, Univ. Roma Tre
DG surface: predicted behavior
● β < βR
β > βR
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Fabio Martinelli, Univ. Roma Tre
High temperature DG vs. the DGFF
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Fabio Martinelli, Univ. Roma Tre
Low temperature DG
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Inhomogeneous Random Systems 2015
Fabio Martinelli, Univ. Roma Tre
Intuition to the BEF’86 results ● 8
Inhomogeneous Random Systems, 2015
Fabio Martinelli, Univ. Roma Tre
Progress for SOS in recent years
[Caputo, Lubetzky, M, Sly, Toninelli ’12, ’13, ’15] :
1
log L ;
• maximum concentrates on
2
1
log L ;
• average height above a floor concentrates on
4
• deterministic scaling limit of the level lines loop ensemble;
• L1/3 fluctuations exponent of the largest loop around its limit.
[Caputo, M, Toninelli ’14] : Large deviations without the floor:
⇡⇤ (⌘⇤ > 0) ⇣ exp( ⌧ L log L)
surface tension
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9
Fabio Martinelli, Univ. Roma Tre
Low temperature DG:
Previous work: [Bricmont, El-­‐Meloukki, Frohlich ’86]: •Maximum XL : E[XL ] ⇣
p
1
log L
•Theorem [Lubetzky,M. Sly]: p
∃ M=M(L) ~ (1/2⇡ ) log L log log L such that XL ∈ {M,M+1} w.h.p.
Remark •for a.e. L (log density) XL=M w.h.p. p
•Extra log
log
L
due to nature of large deviations. •in DG: “harmonic pinnacles” preferables to spikes.
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Fabio Martinelli, Univ. Roma Tre
Low temperature DG
Central ingredient: Large Deviation estimate on ∞-volume DG:
Proposition [Lubetzky, M., Sly] :
⇡(⌘0
h2
h) = exp( (2⇡ + o(1))
)
log h
(cf. exp( ch2 ) for a spike of height h.)
• M:= max integer such that ⇡(⌘0
M
L
2
log5 L)
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Inhomogeneous Random Systems, 2015
Fabio Martinelli, Univ. Roma Tre
Low temperature DG
• Previous work [Bricmont-El Meloukki, Frohlich] : av. height with floor ~
p
1
log L
p
• Theorem [Lubetzky, M, Sly] : conditioned on η > 0 9H ⇠ (1/4⇡ ) log L log log L
such that :
(i) #{x : ⌘x 2 {H, H + 1} }
(1
.
✏( ))L2
(ii) For any h < H : single macroscopic loop with area L2(1-o(1));
(iii) Height H : single macroscopic loop with area (1
✏( ))L2
(iv) No (H+2) macroscopic loop;
(v) No negative macroscopic loops
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Fabio Martinelli, Univ. Roma Tre
Low temperature DG
Roughly put: with the floor, w.h.p. :
• DG surface is a plateau at height H ~ 1/√2 M
• Plateau is approx a raised (by H) version of the surface without floor
• the floor raises the maximum by a factor (1+1/√2).
Theorem [Lubetzky, M, Sly]
Conditioned on η>0 :
1
9M ⇠ (1 + p )M such that, w.h.p. XL 2 {M ⇤ , M ⇤ + 1, M ⇤ + 2}
2
⇤
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Inhomogeneous Random Systems, 2015
Fabio Martinelli, Univ. Roma Tre
Generalizations to p-Hamiltonians
Extensions to surface models with H(⌘) =
X
x⇠y
SOS
⌘y | p ,
|⌘x
DG
RSOS
Example: LD in infinite volume :
2
1
p 2 [1, +1]
log (⇡(⌘o
h)
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Inhomogeneous Random Systems, 2015
Fabio Martinelli, Univ. Roma Tre
From LD in RSOS to the # of alternating sign
matrices (ASMs)
● AMS =
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Fabio Martinelli, Univ. Roma Tre
Height large deviations in the DG
LD dominated by “harmonic pinnacles”, integer approximations to the
Dirichlet problem :
X
Ir (h) = inf{
('x 'y )2 : ' Brc = 0, '0 = h}
x⇠y
• real solution: harmonic function 𝜙:
✓
log |x| + O(1)
x = Px (⌧0 < ⌧@Br )h = 1
log r
P
2
P
(⌧
<
⌧
)
h
x
0
@B
r
Ir (h) = 4h2 x
⇠ 2⇡
E0 (⌧@Br
log r
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◆
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From the real to the discrete Dirichlet problem
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Fabio Martinelli, Univ. Roma Tre
DG with floor
● Main building block : Fix two integers (h, k) such that 4 + 2  k⇡(⌘0
h)  4 + 4
Then, w.h.p. the DG with floor & boundary height (h-­‐1) on a square of side k will contain a h-­‐level loop whose interior fills almost everything. The relation between (h, k) embodies the “entropic repulsion”.
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DG with floor
● Use the building block to recursively raise the surface from height 0 ➞ 1 ➞ 2 ➞ ……. ➞ H where L ⇡(⌘0
H) ⇡ 4
by tiling the original L x L box with boxes of side k = k(h) ⇡
4
⇡(⌘0
h)
,
h = 1, 2, . . .
● An important point is to check that in this process the loss of area near the boundary is negligible. Inhomogeneous Random Systems 2015
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Open problems
● Thank you
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Fabio Martinelli, Univ. Roma Tre