Planar self-avoiding walk scaling limit

Draw a path along a square grid without visiting any vertex twice. After connecting two marked boundary points and making the grid finer, does its random shape approach one continuous-curve law? The question uses critical weights for different path lengths and asks about the shape with the timing of the steps forgotten.[1] [2]

A critical random path that cannot revisit a vertex is conjectured to have a continuum shape law. Is that true for square-grid walks between two fixed boundary points? Approximate a bounded simply connected planar domain with smooth boundary and two distinct marked boundary points. Assign each nearest-neighbor self-avoiding walk of length $\ell$ weight $\mu^{-\ell}$, where $\mu$ is the square-lattice connective constant; normalize over the allowed walks and forget the step parametrization. The target is weak convergence of these traces to chordal $\mathrm{SLE}_{8/3}$ as the mesh vanishes.[1] [2]

There is a precise candidate for the continuous curve, but resemblance in simulations does not prove that the square-grid paths converge to it. Results for the outer edge of an ordinary random walk concern a different object, which can revisit vertices before its frontier is extracted.[1] [2] [3]

The cited candidate argument assumes existence of a conformally covariant scaling limit and the relevant restriction structure. Its authors explicitly leave precise conditional-theorem hypotheses unchecked in that discussion. Proving the discrete-to-continuum passage remains necessary; natural-parametrization convergence of random-walk frontiers does not imply it for self-avoiding walks.[1] [2] [3]

A polymer chain cannot place two of its parts at the same location in this idealized grid model. Self-exclusion makes choices far along the chain depend on earlier steps. A continuum law would explain the resulting large-scale shape without retaining every microscopic grid turn.[1] [2]

A local no-revisit rule imposes long-range dependence, yet its proposed continuum law has rigid conformal structure. This would connect microscopic exclusion to macroscopic curve geometry and explain why continuum calculations predict scaling behavior for planar polymer models. Rigorous properties of the candidate curve alone cannot justify those predictions for the lattice paths.[1] [2]