Dominant-singularity conjecture for adsorbing prudent loops

Let $W(z;1,0;a)$ be the generating function for prudent loops, with step variable $z$ and surface fugacity $a>0$. Let $z_{1}^{\ell}\approx 0.412095$ denote a root of

$$1-3z-z^2+6z^3-7z^7-z^8+3z^9+z^{10}=0,$$

and let $a_{c}^{\ell}\approx 1.82476$ be a root of

$$1-7a+45a^2-143a^3+277a^4-346a^5+285a^6-155a^7+54a^8-11a^9+a^{10}=0.$$

Let $z_{2}^{\ell}(a)=z_{2}^{t}(a)$, where $z_{2}^{t}(a)$ is the root defined by the equation for the adsorbed phase in the corresponding prudent-tail conjecture. Dominant-singularity conjecture. The dominant singularity of $W(z;1,0;a)$ is

$$z^{\ell}(a)=\begin{cases}z_{1}^{\ell}&a\leq a_{c}^{\ell},\\z_{2}^{\ell}(a)&a>a_{c}^{\ell}.\end{cases}$$

Thus the absorbed-state free energy for prudent tails, prudent loops and partially directed walks is determined by the same equation; the adsorption transition is first-order at $a=a_{c}^{\ell}$, with crossover exponent $\phi=1$. The claim is supported by the preceding conjecture and series analysis, but is not proved rigorously in the paper.