The critical-dimension conjecture for alpha-weighted Laplacian loop-erased random walk

Let $\alpha\leq 1$, and let $d$ be the dimension of a $d$-dimensional torus. Consider an $\alpha$-weighted Laplacian random walk, obtained by assigning to each step a weight given by the $\alpha$-th power of the Laplacian weight. Write $d_{\alpha}^{\textrm{crit}}$ for the critical dimension.

Critical-dimension conjecture. The critical dimension is

$$d_{\alpha}^{\textrm{crit}}=\frac{2(1+\alpha)}{\alpha}.$$

Equivalently, for every $d>\frac{2(1+\alpha)}{\alpha}$, the typical path of an $\alpha$-weighted Laplacian random walk on a $d$-dimensional torus has size $N^{d/(1+\alpha)}$, whereas this behavior does not hold for smaller dimensions. The claim extends the complete-graph prediction that the typical path has size $N^{1/(1+\alpha)}$ and identifies the dimension at which the corresponding torus behavior should become mean-field.