Harmonic Pinnacles in the Discrete Gaussian Model
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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 1 The Discrete Gaussian model ● 2 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 3 Inhomogeneous Random Systems, 2015 Fabio Martinelli, Univ. Roma Tre DG surface: basic questions 4 Inhomogeneous Random Systems, 2015 Fabio Martinelli, Univ. Roma Tre DG surface: predicted behavior ● β < βR β > βR 5 Inhomogeneous Random Systems, 2015 Fabio Martinelli, Univ. Roma Tre High temperature DG vs. the DGFF ● 6 Inhomogeneous Random Systems, 2015 Fabio Martinelli, Univ. Roma Tre Low temperature DG ● 7 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 Inhomogeneous Random Systems, 2015 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. 10 Inhomogeneous Random Systems, 2015 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) 11 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 12 Inhomogeneous Random Systems, 2015 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 ⇤ 13 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) Inhomogeneous Random Systems, 2015 14 Fabio Martinelli, Univ. Roma Tre 15 Inhomogeneous Random Systems, 2015 Fabio Martinelli, Univ. Roma Tre From LD in RSOS to the # of alternating sign matrices (ASMs) ● AMS = 16 Inhomogeneous Random Systems, 2015 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 Inhomogeneous Random Systems, 2015 ◆ 17 Fabio Martinelli, Univ. Roma Tre From the real to the discrete Dirichlet problem ● 18 Inhomogeneous Random Systems, 2015 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”. Inhomogeneous Random Systems 2015 Fabio Martinelli Roma Tre 19 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 Fabio Martinelli Roma Tre 20 Open problems ● Thank you 21 Inhomogeneous Random Systems, 2015 Fabio Martinelli, Univ. Roma Tre