Open problem, Probability and statistics, posed 1949

Critical exponents of self-avoiding walks

Open

Let $c_n$ be the number of $n$ -step self-avoiding walks on $\mathbb{Z}^d$ starting at the origin, and let $R_n$ be the root-mean-square end-to-end displacement of a uniformly random such walk. Prove that $c_n \sim A \mu^n n^{\gamma - 1}$ and $R_n \sim B n^{\nu}$ as $n \to \infty$ for critical exponents $\gamma$ and $\nu$ , and determine their exact values in dimensions $d = 2$ and $d = 3$ .

Research frontier as of 2026

As of 2026, the mean-field exponents $\nu = 1/2$ , $\gamma = 1$ are rigorously established for $d \ge 5$ (Hara–Slade 1992), and logarithmic corrections at the upper critical dimension $d = 4$ are rigorously proven via renormalization group methods (Bauerschmidt–Brydges–Slade and collaborators). In $d = 2$ , only the connective constant $\mu = \sqrt{2+\sqrt{2}}$ (Duminil-Copin–Smirnov 2010) is proven exactly; Nienhuis's predicted exponents $\nu = 3/4$ , $\gamma = 43/32$ remain conjectural, tied to unproven conformal invariance of the scaling limit. In $d = 3$ , no exponent has been proven to exist, let alone computed; Monte Carlo estimates give $\nu \approx 0.5876$ , $\gamma \approx 1.1568$ .

Best known results

Mean-field exponents $\nu = 1/2$ and $\gamma = 1$ are proven for $d \ge 5$ via the lace expansion (Hara–Slade 1992), and a logarithmic correction with exponent $1/4$ is proven at $d = 4$ via rigorous renormalization group analysis.
The connective constant on the hexagonal lattice is proven exactly, $\mu = \sqrt{2+\sqrt{2}}$ (Duminil-Copin–Smirnov 2010), but the associated critical exponents $\nu, \gamma$ in $d = 2$ remain conjectural.

Tools and where they stop

Tool	Achieved	Where it stops
Lace expansion	Rigorously proves Gaussian mean-field scaling and critical exponents for $d \ge 5$ by controlling a convergent perturbative series around simple random walk	The expansion coefficients grow too large to sum below $d \approx 5$ , since self-intersections become too frequent in low dimensions for the perturbative series to converge
Conjectural conformal invariance and Schramm–Loewner evolution (SLE $_{8/3}$ )	Predicts the exact 2D exponents $\nu = 3/4$ , $\gamma = 43/32$ and identifies the conjectural scaling limit of the self-avoiding walk as SLE with parameter $\kappa = 8/3$ (Lawler–Schramm–Werner 2004, conditional on conformal invariance)	Conformal invariance of the self-avoiding walk scaling limit is itself unproven, so all consequences (including the exponent values) remain conditional

Open questions

Do the critical exponents $\nu$ and $\gamma$ exist (in the sense of the stated asymptotics) for self-avoiding walk on $\mathbb{Z}^3$ , and if so what are their exact values?
Is the scaling limit of the self-avoiding walk on $\mathbb{Z}^2$ conformally invariant, and does it converge to $\mathrm{SLE}_{8/3}$ ?

References

Neal Madras, Gordon Slade (1993). Random Walks in Random and Non-Random Environments
Takashi Hara, Gordon Slade (1992). Self-avoiding walk in five or more dimensions I: The critical behaviour · DOI:10.1007/BF02096738
Hugo Duminil-Copin, Stanislav Smirnov (2011). Conformal invariance of lattice models · arXiv:1109.1549
Roland Bauerschmidt, David C. Brydges, Gordon Slade (2015). Logarithmic correction for the susceptibility of the 4-dimensional weakly self-avoiding walk: a renormalisation group analysis · arXiv:1403.7422