25 problems

Let $u$ be a probability measure on $mathbb{R}^d$, let $(omegak){k=1}^{infty}$ be independent random points distributed according to $nu$, and for $alpha0$ define … Write…

Let $(\mathcal{T},d_{\mathcal{T}},\mu)$ be the continuum random tree, and for $\sigma\in\mathcal{T}$ and $r>0$ define … The cited global volume estimate gives, almost surely, rando…

Let $K\subset\mathbb{R}$ be a nonrandom compact set, let $\mathfrak S(\theta)$ be the random set of points of slow growth at parameter $\theta$, and let $\lambda(\theta)$ denote th…

Let $B^H=(B_t^H)_{t\in[0,1]}$ be a fractional Brownian motion with Hurst parameter $H\in(0,1)$, and let … be its graph. Fraser–Sahlsten conjecture. Almost surely, … Fraser and Sahl…

Let $\mathbf{c}in(0,1]$ be the intensity of a critical Brownian loop soup, and let $xi_{\mathbf{c}}(2)$ and $xi_{\mathbf{c}}(4)$ denote the generalized disconnection exponents intr…

For each $\underline{\kappa}\in(0,4)$, let $\rho_c=\rho_c(\underline{\kappa})$ lie in $(-2-\underline{\kappa}/2,-2)\cap[\underline{\kappa}/2-4,-2)$, and let $\eta$ be an SLE…

Let $p\in(0,1)$ be the retention parameter in fractal percolation, and let property denote the strong-disconnection property defined in the source. Strong-disconnection conjecture.…

Let $A_\infty$ denote the final retained set in planar fractal percolation, and let $p_c$ be its critical parameter. Critical fractal-percolation carpet conjecture. At the critical…

For $\kappa \in (4,8)$, let $\eta$ be an SLE$_\kappa$ curve from $0$ to $\infty$ in $\mathbb{H}$. The adjacency graph of the connected components of $\mathbb{H} \setminus \eta$ has…

Let the random self-similar set be generated on the line by a homogeneous iterated function system whose common contraction ratio is the reciprocal of a non-negative integer and wh…

Full-dimensionality conjecture. The set of $x\in\mathbb{T}$ satisfying either condition always has full Hausdorff dimension almost surely. This conjecture arises from an informal c…

Positive critical Hausdorff measure conjecture. With probability one with respect to $\mu_{\mathbf{p}}$,

Let the Brownian earthworm's set of holes denote the random set of holes arising in the Brownian earthworm model, and let its Hausdorff dimension be denoted by $\dim_{\mathrm{H}}$.…

Ising cluster-measure conjecture. The cluster measure of the Ising model should be a natural measure on the ${\rm CLE}_3$ carpet.

For $\theta>0$, let $A(\theta)\subset[0,\infty)$ be the scale-invariant random closed set arising from the limiting divisor model, and define … Positivity conjecture. If … then…

Let $\gamma$ be an SLE$_\kappa$ curve in the upper half-plane $\mathbb{H}$, with $kappa i(4,8)$, and let $gamma ibigcapmathbb{R}$ denote its intersection with the boundary. Alberts…

Let $\alpha>59$, let $\gamma\in(0,1)$, and let $N_{\gamma,\alpha}(\lambda)$ be the eigenvalue counting function of the random fractal string in the paper. Let $C(\gamma,\alpha)$ an…

Let $N_{\gamma,\alpha}(\lambda)$ be the eigenvalue counting function for the random fractal strings considered in the paper, with $\alpha\in\mathbb{N}$ and $\gamma\in(0,1)$. In par…

Harmonic-restriction information conjecture. We conjecture that, as soon as $X$ is harmonically nontrivial, such measures encode everything about the restriction of $h$ to $X$ (tha…

Quantum-measure restriction conjecture. We conjecture that $cmu_X$ always determines $h$ restricted to $X$, in the sense that it determines its harmonic extension off $X$.

A uniformly random self-affine set is obtained as the attractor of a random iterated function system of affine contractions, with the randomness specified by the model under consid…

Let $N\geq 2$ and $d\geq 2$, and let $\mathcal C$ be the limiting set of Mandelbrot fractal percolation in $[0,1]^d$ with retention parameter $p$. Define the critical value … where…

Strong open set condition conjecture. This regularity is already guaranteed by the strong open set condition on $\operatorname{int}(J)$.

Strong open set condition conjecture. In the deterministic case, the strong open set condition should imply regularity (condition (iii) of the main theorem).

Let $\operatorname{SLE}(\kappa)$ be the Schramm–Loewner evolution path parameterized by capacity, and let the Hölder exponent obtained by J. Lind be denoted by $\alpha$. Lind's con…