24 problems

Bogomolny–Schmit conjecture. Their nodal lines at large scales should converge to curves described by Schramm–Loewner evolution with parameter $6$, namely $\operatorname{SLE}_6$.

Let $D_n$ denote the maximum interpoint distance, and let the conditions and notation of Theorem apply. The relevant scale is the growth regime in which the quantity governing the…

Let $f$ be a Gaussian field with covariance kernel $r\in L^1(\mathbb{R}^d)$. Suppose the upper bound in Theorem number-count is considered for the corresponding number of high loca…

Let $M$ be a compact smooth $d$-dimensional manifold without boundary, let $K_\lambda$ be the covariance kernel of the Gaussian random wave, and let $\Phi_\lambda$ denote the corre…

Let $q\in L^2(\mathbb{R}^2)$ be an isotropic, nonnegative, nonzero function whose spectral measure has support containing a nonempty open set, and let $f=q\star W$ be the associate…

A Gaussian stochastic field is a stochastic field whose finite-dimensional distributions are Gaussian. Its boundary behaviour means the behaviour of the field when restricted to th…

Let $f$ be a Gaussian random field on $\mathbb{R}^d$, let $K\subset\mathbb{R}^d$ be compact, and let $X$ denote the number of critical points of $f$ contained in $K$. Under mild re…

Let $f$ be a centred stationary Gaussian field in $\mathbb{R}^2$ such that, with probability one, it is sufficiently smooth; the joint distribution of $f$ and its derivatives is no…

Let $\eta_1$ denote the critical one-arm exponent for the planar percolation model generated by the strongly correlated Gaussian field with correlation-decay parameter $\alpha>0$.…

Sarnak's conjecture. There exists $\delta>0$ such that, for every $\ell\in[-\delta,\delta]$, almost surely $\{f=\ell\}\cap\mathbb{R}^3$ contains an unbounded component.

Let $f$ be a planar stationary Gaussian field with covariance function $\kappa$, and suppose quasi-independence means that well-separated percolation events on domains of scale $R$…

Let $f$ satisfy Assumptions and, and suppose there exist $\alpha\in(0,2)$ and $r_0>0$ such that … for all $|x|<r_0$. Let $D\subset\mathbb{R}^2$ be an open rectangle centred at the…

Let $f$ satisfy Assumptions,, and, and let $D\subset\mathbb{R}^2$ be an open rectangle centred at the origin. Write $D_R$ for its dilation by $R$, and let $N_\mathrm{ES}(D_R,\ell)$…

Let $N_\mathrm{LS}(D_R,\ell)$ denote the number of level-set components at level $\ell$ in the dilated domain $D_R$ for the Random Plane Wave. Bogomolny–Schmit variance conjecture.…

Log-correlated Gaussian fields are random fields whose covariance between values at two different sites decays logarithmically with their distance; examples include the two-dimensi…

Let $\beta_{n,k}(u_n)$ be the $k$-th Betti number of the random excursion complex at level $u_n$, and let $\lambda_{n,k}(u_n)$ denote its associated mean-scale parameter. Asymptoti…

Let $T$ be a Gaussian eigenfunction on a compact manifold $\mathbb M$. For $u\in\mathbb R$, define the excursion set … For each Lipschitz–Killing curvature of $A_u(T,\mathbb M)$, c…

Let $\phi$ be the three-dimensional BMS field sampled according to the non-Gaussian measure $\nu_{\phi}$. A field is subordinated to a Gaussian field with finite Wiener chaos repre…

For a Dyson $\beta$-ensemble or an ensemble in another symmetry class, let mesoscopic correlations mean correlations at scales intermediate between the microscopic and macroscopic…

Kahane's half-exponential moment conjecture. A subcritical GMC exists if and only if there is a measure $\mu'\sim\mu$ such that

Let $S$ be a parameter set for a smooth stationary Gaussian field, let $M_S$ denote the maximum of the field over $S$, and let the Steiner formula for the tube around $S$ determine…

Let $S$ be a parameter set for a smooth stationary Gaussian field, and let the Steiner formula for the tube around $S$ provide geometric functionals governing the tail expansion of…

Existence conjecture. It should be possible to construct fields of this type over any manifold $M$, with the resulting Riesz fields understood as solutions of the displayed stochas…

Log-correlated Gaussian fields are Gaussian fields whose correlations decay logarithmically with distance; their extreme-value statistics are expected to resemble those of independ…