MATHPH
In the literature: open
Endpoint-distance exponent of the 3D self-avoiding walk
In plain words
A random path that never visits the same point twice models a polymer chain, and in 3D its size should grow as the number of steps to the power 0.588. Even a power-law improvement over straight-line growth is unproven.
Precise statement
Self-avoiding walk on $Z^3$ with $n$ steps, uniform measure. Prove $\langle \mid w(n)\mid^2\rangle^{1/2} = n^{\nu+o(1)}$ with $\nu\ \text{approximately}\ 0.5876$, or at least $\langle \mid w(n)\mid^2\rangle^{1/2} \le n^{1-\delta}$ for some $\delta > 0$. Answer: a proof.
What would settle it
A polynomial sub-ballistic bound, then existence of $\nu$.
Status in the literature
Sub-ballistic behavior $o(n)$ is proven (Duminil-Copin and Hammond, 2013); in $d = 4$ weakly self-avoiding walk has proven logarithmic corrections (Bauerschmidt, Brydges and Slade).