Almost-sure localization size, critical parameters, and stationary fluctuations for stuck walks

The stuck walk of Erschler, Tóth and Werner (2012) is a one-dimensional nearest-neighbour self-interacting random walk whose next-step bias is a fixed linear combination of the local times on the four edges nearest its current position. For $\alpha\in(\alpha_{L+1},\alpha_L)$, with $\alpha_L=1/(1+2\cos(2\pi/(L+2)))$ decreasing to $1/3$, the walk gets 'stuck': it is eventually confined to $L+2$ sites with positive probability, and the trapped local-time profile converges to an explicit deterministic $\sin^2$-shaped vector. The authors left open whether localization on $L+2$ sites is almost sure (excluding larger traps), what happens at the critical parameters $\alpha=\alpha_{L+1}$, whether the confined chain $(Y_n,\Lambda_n)$ is positive recurrent with an invariant distribution, and how to extend the analysis to wider interaction ranges. Kious (2016) proved that localization on $L+2$ or $L+3$ sites is almost sure, and on exactly 3 sites for $\alpha>1$. This record asks for the completion: exclude the $L+3$ alternative for all $\alpha\in(1/3,1)$, determine the behaviour at the critical parameters, establish positive recurrence and an invariant distribution for the confined chain, and prove precise trapping criteria for finite-range interactions beyond four edges.

Background

Self-interacting random walks are nearest-neighbour walks on the integer lattice $\mathbb{Z}$ whose transition probabilities depend on the whole past trajectory, typically through the local times of the edges or sites already visited; the local time $\ell(n,j)$ of the unoriented edge $j$ (the edge connecting sites $j-1$ and $j$) counts how many times the walk has crossed that edge up to time $n$. The best-studied example is the one-dimensional 'true' self-repelling walk (TSRW), which is pushed towards the less-visited of its two adjacent edges and has a non-trivial scaling limit. Erschler, Tóth and Werner introduced in the paper 'Stuck walks' (Probab. Theory Relat. Fields 154 (2012) 149–163) a one-parameter family that adds an attraction through the next-to-neighbouring edges: with a fixed $\beta>0$ and

$$\Delta(n,j) = -\alpha\,\ell(n,j-1)+\ell(n,j)-\ell(n,j+1)+\alpha\,\ell(n,j+2),$$

the walk started at $X_0=0$ jumps from $X_n$ to $X_n\pm1$ with probability $\exp(\pm\beta\Delta_n)/(\exp(\beta\Delta_n)+\exp(-\beta\Delta_n))$, where $\Delta_n=\Delta(n,X_n)$; a positive 'stream' $\Delta_n$ pushes the walker to the right, a negative one to the left. The choices $\alpha=0$ and $\alpha=-1$ recover the TSRW with edge and with site repulsion; the companion survey 'Some locally self-interacting walks on the integers' by the same authors describes the expected phase diagram across all $\alpha$.

For $\alpha>1/3$ the self-attraction can win and the walk becomes 'stuck'. Let $\mathcal{L}$ denote the (possibly infinite) set of sites visited infinitely often, and define $\alpha_1=\infty$ and $\alpha_L=1/(1+2\cos(2\pi/(L+2)))$ for $L\ge2$, a decreasing sequence with $\alpha_2=1$ and $\alpha_L\downarrow 1/3$. Erschler–Tóth–Werner proved that if $\alpha\in(\alpha_{L+1},\alpha_L)$ then $\mathcal{L}$ consists of $L+2$ consecutive sites with positive probability (an interval with $L$ interior sites and $L+1$ edges), and that on this event the normalized edge-local-time profile converges to an explicit deterministic vector $(u_1,\dots,u_{L+1})$ following a $\sin^2$ curve. The proof combines a combinatorial 'stream control' lemma with a comparison to an auxiliary walk $Y$ confined by force to the interval $[0,L+1]$ with the same transition rule. The phenomenon is reminiscent of the vertex-reinforced random walk (VRRW) on $\mathbb{Z}$ — a walk whose jump probabilities increase with the number of previous visits to the target site, which is also known to localize on finitely many sites — but the mechanism is different: no term here is monotone in the local times, and confinement emerges from the competition between repulsion and attraction.

In their concluding remarks the authors list the questions their first-order methods cannot settle. Kious ('Stuck walks: A conjecture of Erschler, Tóth and Werner', Ann. Probab. 44 (2016) 883–923) took up the most prominent one, reformulating it as the conjecture that for $\alpha\in(\alpha_{L+1},\alpha_L)$ the walk localizes almost surely on exactly $L+2$ sites, and proved it partially: writing $R'$ for the set of sites visited infinitely often, he showed $|R'|\in\{L+2,L+3\}$ almost surely under the same assumptions (Theorem 1.4) — in particular the walk localizes almost surely and traps of $L+4$ or more sites are excluded — and $|R'|=3$ almost surely for $\alpha>1$ (Theorem 1.5). A key tool is a variant of Rubin's construction, a coupling of reinforced random walks with stacks of exponential clocks, adapted to non-monotone effective weight functions. He states the belief that $|R'|=L+3$ 'does not occur, as it corresponds to an unstable limiting behaviour', but leaves it unproved, and explicitly leaves the critical values $\alpha=\alpha_{L+1}$ and everything concerning finer fluctuations untouched. Positive recurrence — the property that a Markov chain returns to a state in finite expected time, which guarantees an invariant (stationary) distribution — of the confined chain $(Y_n,\Lambda_n)$ defined in the problem statement remains conjectural.

Problem Statement

The problem is to complete the rigorous description of the trapping regime of the stuck walk defined above, for every $\alpha>1/3$ including the critical parameters. Four linked questions remain open; together they form the common objective of determining which localization sizes occur almost surely, what happens at criticality, and how the trapped walk fluctuates around its deterministic profile.

1. Localization size. Prove, or disprove, that for every $L\ge2$ and every $\alpha\in(\alpha_{L+1},\alpha_L)$ (equivalently $\alpha\in(1/3,1)$), the walk localizes on exactly $L+2$ sites almost surely; that is, that the event $|R'|=L+3$ has probability zero. (The case $\alpha>1$, where $L=1$, is already proved by Kious.) This is the surviving core of the question of Erschler–Tóth–Werner whether, whenever $\#\mathcal{L}=L+2$ has positive probability, the probability of being eventually trapped on any strictly larger interval is zero.

2. Critical parameters. Determine the behaviour of the walk at $\alpha=\alpha_{L+1}$: prove or disprove the heuristic that the walk is then almost surely not stuck on $L+2$ sites (the boundary streams $d_0$ and $d_{L+1}$ of the limiting profile vanish, so a confined walk should eventually escape), and decide whether the walk can be stuck on $L+3$ sites — or any other finite number of sites — with positive probability at criticality.

3. Fluctuations and stationary measure. For the auxiliary walk $Y$ confined by force to $[0,L+1]$ with the same dynamics (defined, as in the source paper, for $\alpha<\alpha_L$), in the regime where $Y$ visits all sites of the interval infinitely often, prove or disprove that the Markov chain $(Y_n,\Lambda_n)$, where $\Lambda_n=(\ell(n,1),\dots,\ell(n,L+1))-n(u_1,\dots,u_{L+1})$ is the centered local-time profile, is positive recurrent and admits an invariant distribution. This would upgrade the law of large numbers for the trapped profile to a genuine fluctuation theory.

4. Wider interaction range. For the generalized dynamics in which $\Delta(n,j)$ is replaced by a linear combination $\alpha_k\ell(n,j-k)+\cdots+\alpha_1\ell(n,j-1)+\alpha_0\ell(n,j)-\alpha_0\ell(n,j+1)-\cdots-\alpha_k\ell(n,j+1+k)$ involving local times at more than four edges, establish precise criteria on the coefficients (in terms of the roots of the associated polynomial, i.e. of the corresponding generalized Fibonacci recurrence) under which trapping on finitely many sites occurs with positive probability.

The objective is to determine, for every $\alpha>1/3$ including the critical parameters, the localization phase diagram of the model and the fluctuation structure of the confined chain, without restricting the parameter range, the interaction, or the conventions of the model.

References

1. Anna Erschler, Bálint Tóth, Wendelin Werner, 'Stuck walks', Probability Theory and Related Fields 154 (2012) 149–163. DOI: 10.1007/s00440-011-0365-4, https://doi.org/10.1007/s00440-011-0365-4 (also arXiv:1011.1103, https://arxiv.org/abs/1011.1103)
2. Daniel Kious, 'Stuck walks: A conjecture of Erschler, Tóth and Werner', The Annals of Probability 44 (2016) 883–923. DOI: 10.1214/14-AOP991, https://doi.org/10.1214/14-AOP991 (also arXiv:1309.1586, https://arxiv.org/abs/1309.1586)
3. Anna Erschler, Bálint Tóth, Wendelin Werner, 'Some locally self-interacting walks on the integers', in: Probability in Complex Physical Systems, Springer Proceedings in Mathematics 11, Springer (2012) 313–338. DOI: 10.1007/978-3-642-23811-6_13, https://doi.org/10.1007/978-3-642-23811-6_13 (also arXiv:1011.1102, https://arxiv.org/abs/1011.1102)
4. Anne-Laure Basdevant, Bruno Schapira, Arvind Singh, 'Localization of a vertex reinforced random walk on Z with sub-linear weight', Probability Theory and Related Fields 159 (2014) 75–115. DOI: 10.1007/s00440-013-0502-3, https://doi.org/10.1007/s00440-013-0502-3
5. Yuval Peres, Bruno Schapira, Perla Sousi, 'Martingale defocusing and transience of a self-interacting random walk', Annales de l'Institut Henri Poincaré, Probabilités et Statistiques 52 (2016) 1009–1022. DOI: 10.1214/14-AIHP667, https://doi.org/10.1214/14-AIHP667 (also arXiv:1403.1571, https://arxiv.org/abs/1403.1571)
6. Bruno Schapira, 'A 0–1 law for vertex-reinforced random walks on Z with weight of order k^α, α ∈ [0, 1/2)', Electronic Communications in Probability 17 (2012). DOI: 10.1214/ECP.v17-2084, https://doi.org/10.1214/ECP.v17-2084
7. Rosemary J. Harris, 'Random walkers with extreme value memory: modelling the peak-end rule', New Journal of Physics 17 (2015) 053049. DOI: 10.1088/1367-2630/17/5/053049, https://doi.org/10.1088/1367-2630/17/5/053049
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