12 problems

Let the incipient infinite cluster (IIC) be the random configuration obtained as the local limit of critical percolation conditioned on the origin being connected to distance tendi…

A homogeneous space of a compact Lie group is a quotient equipped with the induced smooth structure, and its dimension as a differentiable manifold is its usual manifold dimension.…

Bell–Ho–Strichartz conjecture. As $m\to\infty$, the law of $r\circ c^{(w)}$ under $\lambda_m$ converges to the Dirac measure at $\sqrt{8/75}$, while the law of…

Let bond percolation be defined on a hypercubic lattice, and consider its incipient infinite cluster at criticality. The spectral dimension $d_s$ is defined by the long-time return…

Two-walk intersection conjecture. The factor $\log\log n$ in the numerator of the bound

Self-intersection estimate conjecture. The factor $(\log\log n)^2$ in the numerator of the bound

Self-intersection moment conjecture. There exists a positive constant $C$ such that for every natural number $p$, every $x\in V(G)$, and every $n\geq 3$,

Local spectral dimension conjecture. The local spectral dimension for the non-fixed-point diffusion should be the spectral dimension for the limiting fractal in general.

Let $gamma otin[0,2)$?

Spectral-dimension conjecture. The right-hand side of this ultraviolet expression should always be $D$ for a standard quantum scalar field; equivalently, such a field should not ex…

Let $(\Gamma,\nu)$ be the ag-ensemble, with parameters $\alpha$, $\gamma$, and $D$, and let $X^i_n$ denote the relevant random variable for the ensemble. The ag-ensemble probabilit…

Let $d_{S}$ denote the spectral dimension when it exists, and consider a self-similar space with Hausdorff dimension. Euclidean-cube conjecture. The only self-similar cases in whic…