Math Conjectures

Two-dimensional self-avoiding-walk critical exponents

OPENLandmarkConjectureProposed 1972–1982 · Standard version

Canonical statement

For n≥0n\ge0, let
Sn={ω=(ω0,…,ωn)∈(Z2)n+1:ω0=0, ∥ωj+1−ωj∥1=1 (0≤j<n), ωi≠ωj (0≤i<j≤n)} \mathcal S_n=\{\omega=(\omega_0,\ldots,\omega_n)\in(\mathbb Z^2)^{n+1}:\omega_0=0,\ \|\omega_{j+1}-\omega_j\|_1=1\ (0\le j<n),\ \omega_i\ne\omega_j\ (0\le i<j\le n)\}
and put cn=∣Sn∣c_n=|\mathcal S_n|. Let μ=lim⁡n→∞cn1/n\mu=\lim_{n\to\infty}c_n^{1/n}, whose existence is known, and let Pn\mathbf P_n and En\mathbf E_n denote the uniform probability law on Sn\mathcal S_n and expectation with respect to that law. There exist constants A,D∈(0,∞)A,D\in(0,\infty) such that, as n→∞n\to\infty,
cn∼Aμnn11/32andEn[∥ωn∥22]∼Dn3/2. c_n\sim A\mu^n n^{11/32} \quad\text{and}\quad \mathbf E_n[\|\omega_n\|_2^2]\sim Dn^{3/2}.
Thus the counting and metric critical exponents are respectively γ=43/32\gamma=43/32 and ν=3/4\nu=3/4.
View source LaTeX
For \(n\ge0\), let
\[
\mathcal S_n=\{\omega=(\omega_0,\ldots,\omega_n)\in(\mathbb Z^2)^{n+1}:\omega_0=0,\ \|\omega_{j+1}-\omega_j\|_1=1\ (0\le j<n),\ \omega_i\ne\omega_j\ (0\le i<j\le n)\}
\]
and put \(c_n=|\mathcal S_n|\). Let \(\mu=\lim_{n\to\infty}c_n^{1/n}\), whose existence is known, and let \(\mathbf P_n\) and \(\mathbf E_n\) denote the uniform probability law on \(\mathcal S_n\) and expectation with respect to that law. There exist constants \(A,D\in(0,\infty)\) such that, as \(n\to\infty\),
\[
c_n\sim A\mu^n n^{11/32}
\quad\text{and}\quad
\mathbf E_n[\|\omega_n\|_2^2]\sim Dn^{3/2}.
\]
Thus the counting and metric critical exponents are respectively \(\gamma=43/32\) and \(\nu=3/4\).
PROB-001:[Orr1947SAW][Nienhuis1982Exact][Slade2019SAW]
Probability theoryCombinatoricsMathematical physics

Notes

A self-avoiding walk is a nearest-neighbor lattice path that never revisits a site; the model entered mathematics as a caricature of a long polymer chain in solution [Orr1947SAW]. Let cnc_n be the number of nn-step self-avoiding walks on Z2\mathbb Z^2 starting at the origin. Subadditivity gives a connective constant μ\mu with cn1/n→μc_n^{1/n}\to\mu. The conjecture asserts the finer asymptotics cn∼Aμnn11/32c_n\sim A\mu^n n^{11/32} and, for a uniformly random nn-step walk, mean-square displacement of order n3/2n^{3/2}: the critical exponents are γ=43/32\gamma=43/32 and ν=3/4\nu=3/4. These values emerged from a 1972 polymer-model prediction and were derived exactly, at a physical level of rigor, by Nienhuis in 1982 [Nienhuis1982Exact].

Rigorous results stand in sharp contrast. In high dimensions the lace expansion establishes Gaussian behavior, and much of what is known across all dimensions is collected in Slade's account [Slade2019SAW]. On Z2\mathbb Z^2, however, neither power law is proved, and even the existence of the limits defining γ\gamma and ν\nu is unknown. The predicted SLE8/3\mathrm{SLE}_{8/3} scaling limit would deliver both exponents, but that limit is itself unproved. The conjecture is fully open.

References (3)

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