Hammersley's program for the existence of critical exponents of two-dimensional self-avoiding walks: worm super-multiplicativity, two-layer slab asymptotics, and ratio-of-ratios decay

Guttmann and Jensen (J. Phys. A: Math. Theor. 55 (2022) 415206) made precise three approaches, due to John Hammersley, toward proving that the number $c_n$ of $n$-step self-avoiding walks (SAWs) on the square lattice carries at most a polynomial sub-exponential correction to $\mu^n$ (where $\mu$ is the connective constant) — the key missing step toward a rigorous existence proof of critical exponents for two-dimensional SAWs. Each approach rests on a precise, numerically supported but unproved conjecture: (i) the number $w_n$ of worms — SAWs whose endpoint lies in the same vertical column as the origin — is asymptotically super-multiplicative, which would imply $c_n\le C_1\mu^n n^2$; (ii) for SAWs on a two-layer lattice with fraction $p$ of vertical steps and exponential growth rate $\kappa(p)$, one has $\kappa(p)-\kappa=O(-p\log p)$ as $p\to0$, stated to be necessary and sufficient for $c_n=\exp(\kappa n+O(\log n))$; and (iii) the ratio of alternate ratios $r_n=c_{n+2}c_{n-2}/c_n^2$ satisfies $r_n=1+O(1/n^2)$. All three conjectures remain open.

Background

An $n$-step self-avoiding walk (SAW) on a lattice is a sequence of $n$ nearest-neighbour steps that never revisits a site; let $c_n$ denote the number of $n$-step SAWs on the square lattice $\mathbb{Z}^2$, counted up to translation. Concatenating two SAWs can create a self-intersection, so $c_{n+m}\le c_nc_m$: the sequence is sub-multiplicative, and by Fekete's lemma (sub-additive sequences converge to their infimum) the connective constant $\mu=\lim_n c_n^{1/n}$ exists, with $c_n\ge\mu^n$ as proved by Hammersley in 1961. It is universally expected that $c_n\sim A\mu^n n^{\gamma-1}$ with a critical exponent $\gamma=43/32$ in two dimensions, a value predicted by Coulomb-gas and conformal-field-theory methods and tied to the conjectured Schramm–Loewner evolution (SLE$_{8/3}$), the hypothesised universal random curve describing the walk's large-scale scaling limit; none of this is proved. Since Hammersley and Welsh (1962), the best unconditional upper bound in two dimensions has been $c_n\le\mu^n e^{a\sqrt n}$ for a constant $a$, obtained by the unfolding operation, which reflects segments of a walk to make it monotone in the $x$-direction. Even a bound $c_n\le\mu^n n^{O(1)}$ — the existence of a critical exponent in its weakest quantitative form — has resisted proof for over sixty years. In a 1982 conversation, John Hammersley sketched to Guttmann three purely combinatorial routes to such a bound; Guttmann and Jensen (2022) made these routes precise and tested each numerically.

Route 1: worms. A bridge is a SAW whose origin has the unique minimal $x$-coordinate and whose endpoint has the maximal $x$-coordinate; bridges concatenate without possible self-intersection, so bridge counts are super-multiplicative ($b_nb_m\le b_{n+m}$). A worm is a SAW whose origin and endpoint have the same $x$-coordinate. Any SAW splits, at the last visited vertex of the origin's column and the first visited vertex of the endpoint's column, into two worms separated by a bridge with one appended horizontal step, with at most $n^2$ choices of the two cut vertices. If the worm counts $w_n$ were super-multiplicative — or merely asymptotically so, since low-order failures are harmless — this decomposition yields $c_n\le C_1\mu^n n^2$. Guttmann and Jensen enumerated square-lattice worms to $n=59$ and triangular-lattice worms to $n=40$ (OEIS A356617/A356618) and found super-multiplicativity in every tested square-lattice case with $n+m<60$; moreover the triangular sequence, and the odd and even subsequences of the square sequence, appear log-convex, meaning $w_{n-1}w_{n+1}\ge w_n^2$, which would make the ratios $w_n/w_{n-1}$ an increasing sequence of rigorous lower bounds for $\mu$ (numerically $\mu\ge4.1098$ on the triangular lattice against the estimate $4.15079\ldots$, and $\mu>2.6202$ on the square lattice against $2.6381585\ldots$). A scaling argument (confirmed to five or six digits, and derived independently from Duplantier's polymer-network theory) predicts the worm exponent $\gamma_w=\gamma-\nu=19/32$, where $\nu=3/4$ is the metric exponent.

Route 2: the two-layer slab. Take two horizontal square lattices joined by vertical edges (a height-two slice of $\mathbb{Z}^3$) and let a fraction $p$ of the steps be vertical. Writing $c_n(p)\sim e^{n\kappa(p)+o(n)}$ for the number of such walks, with $\kappa(0)=\kappa=\log\mu$, Guttmann and Jensen state that $c_n=\exp(\kappa n+O(\log n))$ holds if and only if $\kappa(p)-\kappa=O(-p\log p)$ as $p\to0$. In Hammersley's own formulation of the model, a walk segment need only be self-avoiding within each visit to a layer, so the globally self-avoiding walks that Guttmann and Jensen enumerate (by backtracking to $n\le24$, at $p=0.02,0.05,0.1,0.15$) form a proper subset; the model and its delicate analysis trace to Hammersley's 1991 paper, whose proof required a corrigendum. The data show $\kappa(p)-\kappa$ linear in $-p\log p$ with gradient about $+0.27$, that is, $\kappa(p)-\kappa\sim C\,p\log p$ with $C\approx-0.27$.

Route 3: ratio of ratios. Kesten (1963) proved the ratio limit $c_{n+2}/c_n\to\mu^2$ (the index gap 2 avoids the antipodal singularity at $-1/\mu$ that bipartite, so-called loose-packed, lattices introduce for successive ratios), with the convergence rate $O(n^{-1/3})$. Guttmann and Jensen observe that if $r_n=\frac{c_{n+2}c_{n-2}}{c_n^2}$ satisfies $r_n=1+O(1/n^2)$, then $c_n=\exp(\kappa n+O(\log n))$; the condition is also necessary, since $c_n\sim C\mu^n n^g$ forces $r_n\sim1-4g/n^2$. Plots of $r_n-1$ against $1/n^2$ using the 80 known series terms are visually linear, against clear curvature against $1/n$.

For context on the state of the art: Duminil-Copin, Ganguly, Hammond and Manolescu (2020) proved $c_n\le\exp(Cn^{1/2-\epsilon})\mu^n$ for the hexagonal lattice (all $n$) and for $\mathbb{Z}^2$ on a set of $n$ of limit supremum density one — the first improvement of the exponent inside the Hammersley–Welsh exponential; Hutchcroft (2018) proved $c_n\le\exp(o(\sqrt n))\mu^n$ for all $n$ from the sub-ballisticity theorem (which states that a self-avoiding walk typically ends a sub-linear distance from its origin); and Krachun and Panagiotis (2026) proved quantitative sub-ballisticity on the hexagonal lattice. None of these reaches a polynomial or logarithmic correction, and none addresses the worm, slab, or ratio conjectures.

Problem Statement

The problem is to prove, by rigorous combinatorial, probabilistic, or analytic argument, that the sub-exponential correction to the count of two-dimensional self-avoiding walks is at most polynomial — by settling at least one of the three conjectures that Hammersley's program, as made precise by Guttmann and Jensen (2022), leaves open. The three statements share one common objective, the first rigorous polynomial or logarithmic sub-exponential bound on $c_n$; each of them individually implies it, and a proof of $c_n\le C\mu^nn^{O(1)}$ or $c_n=\exp(\kappa n+O(\log n))$ for the square lattice by any other rigorous route equally achieves that objective.

(1) Worms. Let $w_n$ be the number of $n$-step walks on $\mathbb{Z}^2$ that are self-avoiding and whose endpoint has the same $x$-coordinate as the origin. Prove that $(w_n)$ is asymptotically super-multiplicative — concretely, that for some constant $C$ and all $n,m$ with $n+m$ large enough, $w_nw_m\le C\,w_{n+m}$ (finitely many low-order failures are immaterial, as the source explicitly notes) — and thereby complete the Guttmann–Jensen derivation of $c_n\le C_1\mu^nn^2$. A separate closely related open problem, not required to complete this route, is eventual log-convexity: proving that the triangular-lattice worm sequence, and the odd and even subsequences of the square-lattice worm sequence, are eventually log-convex ($w_{n-1}w_{n+1}\ge w_n^2$ for all sufficiently large $n$) would make the successive ratios $w_n/w_{n-1}$ rigorous increasing lower bounds for the connective constant (with the enumerated data, $\mu\ge4.1098$ on the triangular lattice and $\mu>2.6202$ on the square lattice).

(2) Two-layer slab. On the lattice of two horizontal square layers joined by vertical edges, let $\kappa(p)$ be the exponential growth rate of the number of $n$-step self-avoiding walks as a function of the fraction $p$ of vertical steps, so that $\kappa(0)=\kappa=\log\mu$. Prove that

$\kappa(p)-\kappa=O(-p\log p)\quad\text{as }p\to0^+,$

for globally self-avoiding walks on this two-layer slab — the model that Guttmann and Jensen enumerate — and derive from it the bound $c_n=\exp(\kappa n+O(\log n))$ for square-lattice walks. (The source states the $O(-p\log p)$ behaviour to be necessary and sufficient for the $O(\log n)$ bound. Hammersley's original formulation relaxes self-avoidance to within each visit to a layer; the globally self-avoiding walks form a proper subset of that model.)

(3) Ratio of ratios. Prove that

$r_n:=\frac{c_{n+2}c_{n-2}}{c_n^2}=1+O\!\left(\frac{1}{n^2}\right).$

References

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