of $v^{\mathrm{que}}_{p,M}$ and $v^{\mathrm{ann}}_{p,M}$ as functions of $M$ for a fixed $p\in(0,1)$. By (\[op1\]), $v^{\mathrm{que}}_{p,M},v^{\mathrm{ann}}_{p,M}\sim \sqrt{2pM}$ as $M\to 0$. []{data-label="fig"}](op2.eps){height="7cm"}

Is this really the case? These questions are presumably hard to resolve analytically. We are not aware even of any simulations done on this problem (see though [@MG02] as well as [@GP82]).

Organization of the paper.
--------------------------

In Section  2 we prove Theorem \[vque\]. The proof of Theorem \[vann\] is given in Section 3, which is subdivided into 3 subsections. Subsection 3.1 gives heuristics and an outline of the proof. Subsection 3.2 is the technical core of the proof. There, after providing an informal calculation, we study the environment under the annealed measure. Subsection 3.3 uses the estimates obtained in the previous subsection and completes the proof of Theorem \[vann\]. The Appendix contains several auxiliary results and proofs of several lemmas used in the main part of the paper.

Terminology.
------------

Sites at which the potential is equal to $M$ will be called [*occupied* ]{} sites or [*obstacles*]{}. [*Vacant*]{} sites are unoccupied sites. An interval is [*empty*]{} if all its sites are unoccupied. We reserve the term [*vacant interval*]{} for maximal empty intervals. A [*gap*]{} between two occupied sites is 