Existence and value of the closing exponent for self-avoiding walks in low dimensions

For the uniform self-avoiding walk of length n on the integer lattice $\mathbb{Z}^d$, the closing probability $q_n$ that the walk's endpoint neighbours its starting point is predicted to decay as $n^{-\psi+o(1)}$ with $\psi=59/32$ in $d=2$, via the scaling relation $\psi=\theta+\xi-1$ between the polygon exponent $\theta$, the walk exponent $\xi$, and the closing exponent. Hammond's snake-method paper (Trans. Amer. Math. Soc. 372, 2019, reference 1) proved the general-dimension upper bound $q_n\le n^{-1/2+o(1)}$ and recorded that the existence and values of these exponents remain open in low dimensions. The problem of rigorously establishing the existence and value of the closing exponent in $d=2$ and $d=3$ is open.

Background

A self-avoiding walk (SAW) of length $n$ on the integer lattice $\mathbb{Z}^d$ is a path $\gamma:\{0,\dots,n\}\to\mathbb{Z}^d$ taking nearest-neighbour steps that never revisits a vertex; it is the standard lattice model of a linear polymer. Writing $c_n$ for the number of such walks starting at the origin, the limit $\mu=\lim_n c_n^{1/n}$, called the connective constant, exists by a submultiplicativity argument of Hammersley, and Kesten (1963) established the corresponding ratio-limit behaviour. A self-avoiding polygon (SAP) of length $n$ is a self-avoiding walk of length $n-1$ whose endpoint neighbours the start, closed by one further step; $p_n$ counts such polygons up to translation. If $\Gamma$ is chosen uniformly among length-$n$ walks, the closing probability $q_n=\mathsf{W}_n(\Gamma\text{ closes})$ — the probability that the endpoint is a nearest neighbour of the origin — satisfies the exact identity

$$q_n=\frac{2(n+1)p_{n+1}}{c_n},$$

so closing-probability decay and polygon/walk enumeration are two views of one question.

Physics predicts power laws: $p_n=n^{-\theta+o(1)}\mu^n$ for even $n$ and $c_n=n^{\xi+o(1)}\mu^n$, whence $q_n=n^{-\psi+o(1)}$ with the closing exponent $\psi=\theta+\xi-1$. In $d=2$, Nienhuis's Coulomb-gas prediction (1982) — obtained by mapping the planar model onto an exactly solvable two-dimensional Coulomb gas — gives $\theta=5/2$ and $\xi=11/32$, so $\psi=59/32$ is predicted; the same values follow from conformal-field-theory heuristics. Rigorous knowledge is sharply dimension-dependent: for $d\ge5$, above the upper critical dimension (the dimension above which self-avoidance becomes a weak perturbation and Gaussian mean-field behaviour holds), the lace expansion (a perturbative expansion exploiting the weak effect of self-avoidance) yields Gaussian scaling and the exponents (Hara–Slade, 1992), and on the hexagonal lattice the connective constant is known exactly (Duminil-Copin–Smirnov, 2012); but in $d=2,3,4$ no critical exponent for the self-avoiding walk has ever been proved to exist.

The best rigorous closing-probability bounds in low dimension come from the snake method, a probabilistic technique introduced by Duminil-Copin, Glazman, Hammond and Manolescu (2016). It decomposes a closing walk into two parts at the walk's lexicographically maximal vertex and exploits Kesten's pattern theorem — which guarantees that any fixed finite walk pattern appears linearly often in all but exponentially few long self-avoiding walks — together with the Gaussian fluctuation of pattern counts on the scale $\sqrt{n}$. That work gave $q_n\le n^{-1/4+o(1)}$ in general dimension and proved that the walk is sub-ballistic, meaning its typical displacement grows slower than linearly in $n$. Hammond's paper (Trans. Amer. Math. Soc. 372, 2019, reference 1) reworked the method to reach $q_n\le n^{-1/2+\varepsilon}$, for any $\varepsilon>0$, for every $d\ge2$ and all large odd $n$; its companion papers strengthened this to $q_n\le n^{-4/7+o(1)}$ along a positive-density set of odd $n$ in $d=2$ (via polygon joining) and to $p_n\le n^{-3/2+o(1)}\mu^n$ along even $n$ of full density in $d=2$. That paper notes that the method could not exceed the $n^{-1/2}$ barrier, that the stronger decay rates are confined to subsequences or to the plane, and that the predicted value $\psi=59/32$ lies far beyond all current bounds.

Problem Statement

Determine, rigorously, the power-law decay of the closing probability of the uniform self-avoiding walk on the nearest-neighbour lattice $\mathbb{Z}^d$ in the low dimensions $d=2$ and $d=3$.

1. (Existence) Prove, or disprove, that there exist exponents $\theta$ and $\xi$ such that $p_n=n^{-\theta+o(1)}\mu^n$ (for even $n$) and $c_n=n^{\xi+o(1)}\mu^n$, and hence — via the exact identity $q_n=2(n+1)p_{n+1}/c_n$ — that the closing probability satisfies $q_n=n^{-\psi+o(1)}$ with the closing exponent $\psi=\theta+\xi-1$.
2. (Value) If the exponents exist, determine their values; in particular, prove or refute the $d=2$ prediction $\psi=59/32$ (equivalently $\theta=5/2$ and $\xi=11/32$).

A rigorous proof that the exponent fails to exist in $d=2$ or $d=3$ equally resolves the question by refuting the predicted form. Any resolution must hold along the full sequence of lengths $n$ (for $q_n$, all odd $n$ — lattice parity forces $q_n=0$ for even $n$ on $\mathbb{Z}^d$), not merely along a subsequence.

References

1. Alan Hammond, On self-avoiding polygons and walks: The snake method via pattern fluctuation, Transactions of the American Mathematical Society 372 (2019) 2335–2356, DOI 10.1090/tran/7494 (arXiv:1808.09597), https://doi.org/10.1090/tran/7494
2. Alan Hammond, On self-avoiding polygons and walks: the snake method via polygon joining, Electronic Journal of Probability 24 (2019), Paper No. 49, 43 pp., DOI 10.1214/18-EJP249 (arXiv:1808.10500), https://doi.org/10.1214/18-EJP249
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