We study the low temperature (2+1)D Solid-On-Solid model on [[1,L]]2
with zero boundary conditions and non-negative heights (a floor at height 0).
Caputo et al. (2016) established that this random surface typically admits
either h or h+1 many nested macroscopic level line
loops {Liβ}iβ₯0β for an explicit hβlogL,
and its top loop L0β has cube-root fluctuations: e.g., if Ο(x)
is the vertical displacement of L0β from the bottom boundary point
(x,0), then maxΟ(x)=L1/3+o(1) over xβI0β:=L/2+[[βL2/3,L2/3]]. It is believed that rescaling Ο by
L1/3 and I0β by L2/3 would yield a limit law of a diffusion on
[β1,1]. However, no nontrivial lower bound was known on Ο(x) for a fixed
xβI0β (e.g., x=2Lβ), let alone on minΟ(x) in I0β, to
complement the bound on maxΟ(x). Here we show a lower bound of the
predicted order L1/3: for every Ο΅>0 there exists Ξ΄>0 such
that minxβI0ββΟ(x)β₯Ξ΄L1/3 with probability at least
1βΟ΅. The proof relies on the Ornstein--Zernike machinery due to
Campanino--Ioffe--Velenik, and a result of Ioffe, Shlosman and Toninelli (2015)
that rules out pinning in Ising polymers with modified interactions along the
boundary. En route, we refine the latter result into a Brownian excursion limit
law, which may be of independent interest.Comment: 48 pages, 2 figure