Stuck walks: a conjecture of Erschler, Tóth and Werner (Q282489)

Let \(X=\{ X_n\}_{n=0}^\infty\) be a nearest neighbor random walk on the integer lattice \({\mathcal Z}\) and let \(l_k(j)\) denote the local time on the nonoriented edge \(\{j-1,j\}\). Next, for a real parameter \(\alpha\) a linear combination \(\Delta_k(j)\) of local times on neighboring and next-to-neighboring edges of a site \(j\) is defined as NEWLINE\[NEWLINE\Delta_k(j)=-\alpha l_k(j-1)+l_k(j)-l_k(j+1)+\alpha l_k(j+2).NEWLINE\]NEWLINE The paper studies the random walk introduced by \textit{A. Erschler} et al. [Probab. Theory Relat. Fields 154, No. 1--2, 149--163 (2012; Zbl 1261.60096)]. The random walk in that paper is defined by \(X_0=0\) and the conditional transition probabilities NEWLINE\[NEWLINE\operatorname{P}(X_{k+1}=X_k\pm 1| {\mathcal F})=\frac{e^{\pm \beta\Delta_k}}{e^{- \beta \Delta_k}+ e^{ \beta\Delta_k}},NEWLINE\]NEWLINE where \(\beta>0\) and \({\mathcal F}_k\) is the filtration of the process. Define the sequence \(\{\alpha_L\}\) by NEWLINE\[NEWLINE\alpha_1=+\infty,\;\alpha_L=\frac{1}{1+ 2\cos(2\pi/(L+2))},\;L\geq 2.NEWLINE\]NEWLINE Denoting by \(Z_k(j)\) the number of visits to the site \(j\) up to time \(k\) and \(R^\prime=\{j\in {\mathcal Z}: Z_\infty(j)=\infty \}\), it was conjectured in [loc. cit.] that if \(\alpha\in (\alpha_{L+1},\alpha_L)\), then \(R^\prime=L+2\) a.s.NEWLINENEWLINEThe author of the present paper gives a partial proof of the conjecture by showing that \(R^\prime =\{L+2,L+3\}\) a.\,s.