Making the lattice-point integral representation for self-avoiding walks on $\mathbb{Z}^d$ effective: rigorous main-term asymptotics and error control for $c_n$

A 2025 preprint of Lazar reduces the counting of self-avoiding walks (SAWs) of length $n$ on $\mathbb{Z}^d$ to counting integer points in a high-dimensional domain and, via Poisson summation, to an explicit oscillatory integral over $\mathbb{R}^{n(n-2)(n+1)}$ whose integrand is a product of sinc and Bessel $J_1$ factors. The paper leaves this representation ineffective: the leading $\nu=0$ integral $I_n(0)$ is not evaluated, no error bound of the form $c_n = \text{main term} + O(\cdot)$ is proved, the estimates are not uniform in the lattice dimension $d$, and the companion second-moment representation is not converted into any bound on the mean-squared displacement beyond the known $o(n^2)$. The open problem is to make this representation analytically effective: rigorously justify the Poisson-summation step, asymptotically evaluate the main term, and control the remainder — yielding the first effective asymptotic formula or two-sided bounds for $c_n(d)$ on $\mathbb{Z}^d$ — with tracing the constants in $d$ and obtaining any quantitative improvement of the displacement exponent as companion directions stated in the source.

Background

A self-avoiding walk (SAW) of length $n$ on the lattice $\mathbb{Z}^d$ is a nearest-neighbor path $\omega = (w_0,\dots,w_n)$ starting at the origin that never revisits a site ($w_j \neq w_k$ for $j \neq k$); $c_n(d)$ denotes the number of such walks. SAWs were introduced by Orr and Flory around 1940 as models of long polymer chains. By Hammersley's submultiplicativity argument, $c_n(d)^{1/n}$ converges to a constant $\mu_d$, the connective constant of the lattice. Classical facts frame the difficulty: Hammersley and Welsh (1962) proved the best general upper bound $c_n \leq \mu_d^n e^{C\sqrt{n}}$; Kesten (1963) proved that the two-step ratio $c_{n+2}/c_n$ converges to $\mu_d^2$, but the existence of the one-step limit $\lim c_{n+1}/c_n$ remains open in full generality; the exact value of $\mu_d$ is unknown for the square lattice ($d=2$), the one famous exception being the hexagonal lattice where Duminil-Copin and Smirnov (2012) proved Nienhuis's prediction $\mu = \sqrt{2+\sqrt{2}}$. Physics predicts (Nienhuis 1982, via the $O(n)$ model — a family of lattice loop models indexed by a loop weight $n$, from which SAWs emerge in the $n \to 0$ limit —) the refined asymptotic $c_n \sim A n^{11/32} \mu^n$ in the plane, and (Flory) that, in the plane, the mean-squared displacement $\langle \|w_n\|^2\rangle$ of the endpoint of a typical SAW grows as $n^{3/2+o(1)}$; rigorously, the best known bound is only $\langle \|w_n\|^2\rangle = o(n^2)$, due to Duminil-Copin and Hammond (2013).

Lazar (2025) proposes a new formalism. Encoding a walk by its step increments $x \in \{0,\pm 1\}^{2n}$, self-avoidance becomes the condition $1 \leq \|w_k - w_j\| \leq k-j$ for all $0 \leq j < k \leq n$, so $c_n$ counts lattice points of $\mathbb{Z}^{2n}$ in the hypercube $[-1,1]^{2n}$ carved by ellipsoidal shells. After a unimodular change of variables (a linear change of coordinates with integer entries and determinant $\pm 1$) and a mollification (replacement of the discontinuous indicator by a smooth approximation), the Poisson summation formula (which relates a sum of a function over the integer lattice to a sum of its Fourier transform) converts this count into an explicit integral representation: $c_n(d)$ is, in the weak sense, a sum over frequencies $\nu \in \mathbb{Z}^{dn}$ of oscillatory integrals $I_n(\nu)$ (for $d = 2$ each $I_n(\nu)$ runs over $\mathbb{R}^{n(n-2)(n+1)}$) whose integrand is a product of roughly $n^2/2$ factors built from the sinc function ($\mathrm{sinc}(x) = \sin(\pi x)/(\pi x)$) and Bessel functions $J_{d/2}$ arising as Fourier transforms of intervals and of balls (Theorem 1.1 for $d=2$, Theorem 1.2 for general $d \geq 2$). The $\nu = 0$ term is expected to dominate, and the final remarks observe that the multidimensional integral is 'essentially of Gaussian type', reducing morally to determinants of hypermatrices, which the author hopes to estimate in any dimension. Nothing of this program is carried out: the $\sim$ statements are formal (derived 'in the weak sense', with no error terms), the leading integral $I_n(0)$ is left unevaluated, and the paper explicitly notes that only the main term, not the error term, is being isolated.

Problem Statement

Make the Poisson-summation lattice-point representation of Lazar (arXiv:2504.13093) for self-avoiding walks on $\mathbb{Z}^d$ analytically effective. Concretely, for at least one fixed dimension $d \geq 2$ (the square lattice $d = 2$ suffices), the required overarching objective is a rigorous asymptotic formula for $c_n(d)$ derived from this representation, comprising: (i) a mathematically complete justification of the integral representation itself, i.e. of the relation $c_n(d) \sim \sum_{\nu \in \mathbb{Z}^{dn}} I_n(\nu)$ with the sinc–Bessel integrands of the source, including the mollification and Poisson-summation steps and their boundary contributions; (ii) an asymptotic evaluation of the leading term $I_n(0)$ (or of the truncated integral $I_n^{\delta}$ for an explicit $\delta$), identifying the dominant growth of $c_n(d)$; and (iii) a quantified remainder bound of the form $c_n(d) = \text{main term} + O(\cdot)$, i.e. control of the Poisson error terms $\sum_{\nu \neq 0} I_n(\nu)$ and of the truncation error. Two companion directions of the same effectiveness program, stated as such in the source, are: (iv) estimates for the multidimensional integrals and hypermatrix determinants that are uniform in the lattice dimension $d$, making the $d \geq 2$ representation effective for $c_n(d)$; and (v) the analogous effective treatment of the second-moment convolution ratio, yielding any rigorous quantitative improvement of the bound $\langle \|w_n\|^2\rangle = o(n^2)$ on $\mathbb{Z}^d$ (the full Flory prediction in the plane being $n^{3/2+o(1)}$). The classical enumeration problems that motivate the source — the existence of $\lim_{n} c_{n+1}/c_n = \mu$ in full generality and the exact value of the connective constant of the square lattice — remain open, but the effective asymptotics of (i)–(iii) do not require resolving those famous problems.

References

1. Youssef Lazar (2025), 'A lattice point counting approach for the study of the number of self-avoiding walks on $\mathbb{Z}^d$', arXiv:2504.13093 (preprint), DOI 10.48550/arxiv.2504.13093, https://arxiv.org/abs/2504.13093
2. Harry Kesten (1963), 'On the Number of Self-Avoiding Walks', Journal of Mathematical Physics 4(7), 960-969, DOI 10.1063/1.1704022, https://doi.org/10.1063/1.1704022
3. J. M. Hammersley and D. J. A. Welsh (1962), 'Further results on the rate of convergence to the connective constant of the hypercubical lattice', The Quarterly Journal of Mathematics 13(1), 108-110, DOI 10.1093/qmath/13.1.108, https://doi.org/10.1093/qmath/13.1.108
4. Hugo Duminil-Copin and Stanislav Smirnov (2012), 'The connective constant of the honeycomb lattice equals $\sqrt{2+\sqrt{2}}$', Annals of Mathematics 175(3), 1653-1665, DOI 10.4007/annals.2012.175.3.14, https://doi.org/10.4007/annals.2012.175.3.14
5. Hugo Duminil-Copin and Alan Hammond (2013), 'Self-Avoiding Walk is Sub-Ballistic', Communications in Mathematical Physics 324(2), 401-423, DOI 10.1007/s00220-013-1811-1, https://doi.org/10.1007/s00220-013-1811-1
6. Bernard Nienhuis (1982), 'Exact Critical Point and Critical Exponents of $O(n)$ Models in Two Dimensions', Physical Review Letters 49(15), 1062-1065, DOI 10.1103/PhysRevLett.49.1062, https://doi.org/10.1103/PhysRevLett.49.1062
7. Dmitrii Krachun and Christoforos Panagiotis (2023), 'Quantitative sub-ballisticity of self-avoiding walk on the hexagonal lattice', arXiv:2310.17299, https://arxiv.org/abs/2310.17299