89 problems

Let $E$ be an eigenvalue of the Laplacian on the $d$-dimensional torus, and let $\mathcal{N}$ be the dimension of the eigenspace corresponding to $E$. Consider arithmetic random wa…

Let $x\in \mathcal{S}^{2}$ and, for $t\in \mathbb{R}$, let $\mathcal{L}_{l;x}^{t}$ denote the random length of the unique component of $f_l^{-1}(t)$ that contains $x$ inside, or…

Refined asymptotic conjecture. Under further assumptions on the sequence $(\lambda(i))$, one can prove that

Markov field property. The local causal states form a Markov field in space-time.

High-level universality conjecture. The high-level regime exhibits the same qualitative behavior as in the monochromatic case.

Mixed Poisson–FGF conjecture. The limiting critical point measure should exhibit mixed behavior, combining a Poisson component and an independent $\mathrm{FGF}_{1/2}(\mathbb{R}^d)$…

Normality and convolution conjecture. The limit law for $f-E(f\mid \mathcal I)$ is normal, and the limit law for $f$ is a convolution of the limit laws for $E(f\mid \mathcal I)$ an…

Non-degeneracy conjecture. The limiting fields are non-degenerate only for the values of the scaling parameters $rho_1$, $rho_2$, and $rho_3$ specified in the paper.

Let $\mathcal{M}$ be a chaotic smooth compact $d$-manifold, and let $\{(\varphi_{j},\lambda_{j})\}_{j\ge 1}$ be the corresponding sequence of Laplace eigenfunctions and eigenvalues…

Multiplicative-noise extension conjecture. Theorem (ii) and (iii) for the additive-noise equation remain valid in the presence of multiplicative noise, provided that

Uniform dimension conjecture. The uniform dimension theorem stated for $p\geqslant4$ is valid whenever $p\geqslant2$, provided that

Let $X$ and $W$ be spatial Gaussian random fields on ${ R} ^d$ with smoothness parameters $\nu_X$, $\nu_W$, and cross-smoothness parameter $\nu_{XW}$. Assume that $\nu_X$ is not a…

Let $X$ be an exposure process, $W$ an unmeasured spatial confounder, and $Y$ a response whose regression slope on $X$ is $\beta$. Say that $W$ is smoother than $X$ when its smooth…

Let the KPZ fixed point have a general initial condition that grows sufficiently more slowly than $f(x)=x^2$ as $|x|$ becomes large, so that the KPZ fixed point exists at $(0,1)$.…

Let $\mathcal{H}^{\mathrm{UT}}$ and $\mathcal{H}^{\mathrm{UT}}_0$ be the upper tail fields of the KPZ fixed point. Upper-tail-field initial-condition independence conjecture. The f…

Let $\mathcal{H}^{\mathrm{UT}}$ and $\mathcal{H}^{\mathrm{UT}}_0$ denote the upper tail fields of the KPZ fixed point obtained near a conditioned high point. Upper-tail-field perio…

The KPZ fixed point is a $(1+1)$-dimensional space-time random field associated with interface-growth models in the Kardar–Parisi–Zhang universality class. KPZ fixed-point universa…

Let $f$ be the spin random field under consideration, let $\alpha0\mathcal{L}_1^f$ and $\alpha0\mathcal{L}_2^f$ denote the corresponding Lipschitz–Killing curvatures, let…

Randomly shifted Gumbel convergence conjecture. There exists a random variable $Z>0$ such that

Conjecture on RMod extensions. It should also be possible to construct and study RMod series in several different settings, including multivariate RMod series and RMod random field…

The geodesic curves of the random fields dynamics are considered on a parametric space whose curvature may vary along their trajectories; the trajectories are described by the Eule…

Let $T_{\ell;d}$ be random hyperspherical harmonics on $\mathbb S^d$, and let the nodal volume be studied in the high-energy limit $\ell\to+\infty$. Gaussian-fluctuations conjectur…

Let $T_{\ell;d}$ be random hyperspherical harmonics on $\mathbb S^d$, and consider the chaotic expansion of their nodal volume into its components. For $d=2$, the fourth chaotic co…

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 $\psi_t$ be the random field obtained by evolving random initial data under the discrete nonlinear Schrödinger equation, and let $\lambda\neq 0$ be the nonlinearity parameter.…