24 problems

For $\gamma\in(0,2)$, consider the $\gamma$-Liouville quantum gravity ($\gamma$-LQG) sphere equipped with its associated metric, whose construction is known in the generality descr…

Let $D_\Phi$ be the $\gamma$-Liouville quantum gravity metric on a $\gamma$-LQG surface, and let $\zeta$ denote its Weyl scaling exponent. Weyl-scaling exponent conjecture. The exp…

Set $\alpha \defeq 1/2+4/\kappa$. Let $(\mathcal{D},\Phi,0,\infty)$ be a $\gamma$-Liouville quantum gravity disk, and let $\Gamma$ be an independent non-nested…

Let $X$ be the carpet of a non-nested $\operatorname{CLE}_\kappa$. Given $X$, let $x$ and $y$ be independent samples from the canonical conformally covariant probability measure…

Let $X$ be the carpet of a non-nested $\operatorname{CLE}_\kappa$. Given $X$, let $x$ and $y$ be independent samples from the canonical conformally covariant measure $\mu_X$, norma…

For $\kappa \in (8/3,4)$, let $\theta(\kappa)$ denote the distance exponent associated with the canonical geodesic metric on the $\operatorname{CLE}_\kappa$ carpet. Distance-expone…

Let $\kappa \in (8/3,4)$, and let a discrete loop model converge to $\operatorname{CLE}_\kappa$ in the scaling limit. Let the chemical distance metric be the graph metric associate…

Let $D_n(\cdot,\cdot)$ be the exponential metric, normalized by the median of the set-to-set distance, and suppose that a limiting metric exists in the phase where…

Kendall's conjecture. Almost surely, in the planar case $d=2$, the speed of every geodesic is bounded away from zero away from its endpoints; equivalently, the non-trivial road seg…

The Calabi metric on the space of Kähler metrics in a fixed Kähler class is the restriction of the deWitt–Ebin metric, and its finite-dimensional approximations are domains in fini…

Let $T(0,v)$ be the passage time from $0$ to $v$, let $T_t(0,v)$ denote its dynamical version at time $t$, and let $\pi_t(0,v)$ be a geodesic from $0$ to $v$ at time $t$. Write…

The directed landscape $\mathcal{L}$ is the full scaling limit of a two-dimensional random metric arising from Brownian last passage percolation and other exactly solvable models i…

Supercritical LQG metric convergence conjecture. The graph distance in the adjacency graph of squares of $\mathcal S_h^\varepsilon$, suitably re-scaled, converges to the metric ass…

Geodesic wandering exponent conjecture. It is conjectured that

KPZ variance exponent conjecture. It is conjectured that

Let $\Pi$ be a planar Poisson line process generating a SIRSN, and let a $\Pi$-geodesic be a fastest route between two specified points. A complete halt means that the route stops…

Let $Q>0$ and let $\xi=\xi(Q)>0$ be related to $Q$ as in the preceding discussion, so that the typical LFPP and square-subdivision distance exponents satisfy the stated relation. L…

Loop-cluster metric conjecture. There is a metric on the clusters of $\mathcal{L}_{g,1/2}$ inside $\Sigma$, measurable with respect to $\mathcal{L}_{g,1/2}$, obtained as the limit…

Let $\theta$ denote the Liouville quantum gravity field and let $u$ be a random metric space whose metric-ball growth process is represented by the random growth process construct…

Exponential convergence conjecture. We conjecture that the convergence to the stationary measure is exponential when $d_\Gamma(I,O)\geq 2$.

Tight-fluctuation conjecture. The centered fluctuations $D(n)-\mathbf{E}[D(n)]$ are tight as $n\rightarrow\infty$, without rescaling.

Consider a minimizing geodesic from $0$ to $rw\in\mathbb R^d$, where $r>0$ and $w$ is a unit direction. Its transverse fluctuations from the straight line segment joining the endpo…

Let $B_g(t)$ be the random metric ball at time $t$, and let $B(t/\mu)$ be the corresponding rescaled limiting shape, where $\mu$ is the time constant. The fluctuations of these two…

Variance-scaling conjecture. The variance of the time constant should behave like