conclude that the expected gap size is $$\frac{1}{K}\sum_{r=1}^\infty(e^\beta(1-p))^r=\frac{1}{K}\,\sum_{r=1}^\infty(1-e^{-K})^r=\frac{e^K-1}{K}.$$ This immediately leads to (\[gap\]).

Here is a layout of the rest of this subsection. We start a rigorous analysis by noticing that all information about the dependence of (\[23\]) on $(r_1,r_2,\dots,r_n)$ is contained in the last product. To study its behavior, we consider an auxiliary quantity, the probability $U_n(q)$ that the killed random walk reaches the $n$-th occupied site prior to the first return to $0$ if $(R_i)_{i\in{\mathbb{N}}}$, are i.i.d. positive integer-valued random variables with probability mass function $G_q(r):=q(1-q)^{r-1}$, $r\in{\mathbb{N}}\cup\{0\}$, for some $q\in (0,1)$. Without loss of generality we shall assume that $0$ is occupied. Then (setting $r_0=0$, $u_0=0$) $$\label{Un}
  U_n(q)=\sum_{r_1,r_2,\dots,r_n\in{\mathbb{N}}} \prod_{i=1}^n G_q(r_i)
F_M(r_{i-1}, r_i, u_{i-1}).$$ We notice that $U_n(q)$ decays exponentially fast in $n$ for each $q$ (Lemma \[lem5basic\]). If we want the event that the killed random walk reaches the $n$-th occupied site prior to the first return to $0$ to be a typical event, then we need to renormalize (\[Un\]). In Corollary \[corbase\] we show that there is $q=q(p)$ such that, after the renormalization, the probability of the above event is essentially equal to 1 (see (\[co1\])). The renormalized measures (\[pi\]) can be effectively compared with product measures 