15 problems

Let $f_{\text{\sc ise}}$ be the density of the integrated super-Brownian excursion (ISE), whose existence and Hölder continuity are established in the paper. ISE density differenti…

Let $N=\sum_{m=2}^{n}I_{[\tau_m>0]}$ be the number of states visited by the process. Visited-state conjecture. In an earlier version of the manuscript, the authors conjectured that…

Let $w,w'$ be two words, and let $\tilde{c}(w)$ denote the second auto-correlation constant defined by the asymptotic second moment of the number of leaves associated with $w$. The…

Total variation separation conjecture. For every such pair,

Let $(X(k))_{k \geqslant 1}$ be a sequence of independent random variables uniform on $\{\pm 1\}$, and define $(A(N))_{N \geqslant 0}$ by the paper's formal power series. For any f…

Let $(A(N))_{N \geqslant 0}$ be defined by the same formal power series from a sequence $(X(k))_{k \geqslant 1}$. Real-Gaussian and Rademacher extension conjecture. The first-momen…

Let $(A(N))_{N \geqslant 0}$ be defined by the formal power series associated with a sequence $(X(k))_{k \geqslant 1}$ of independent standard complex Gaussians. First-moment asymp…

For each of the aforementioned models (1)–(3) of Section in the paper, let $f_N$ be the corresponding random function on $\bar\Lambda_N$, and define its rescaling on $[0,1]^2$ by ……

Let $K_1$, $K_2$, and $k$ be random compact subsets of $\mathbb C$ with $K_1(\omega)\cap K_2(\omega)=\emptyset$ for every $\omega$. Assume … Let…

Let $\mu$ be a finite measure with $r_\mu=1$, let $p>0$, and let $\bm{F}(z)=\sum_{n=0}^\infty a_n\zeta_n z^n$ be a Gaussian analytic function as above. Write $A^{p/2}(\mu)$ for the…

Hall et al.'s conjecture. In the general case, obtaining a non-negligible advantage requires

Convergence conjecture. The random functions $Y_n$ themselves converge, and their limit is not a fixed function.

Let $\mathop{\mathrm{Crt}}\nolimits_{N,0}(-\infty,u)$ denote the number of index-zero critical points of the Hamiltonian below level $u$. The quenched–averaged complexity conjectur…

Necessity conjecture. The assumption that, for given $s\geq 2$, there exists a universal constant $c(s)$ with $\gamma<c(s)$ is necessary to obtain the results presented in Theorem…

Let $S_N(x,t)$ be a time-dependent Steinhaus series and let $\mathcal{E}_N(x,t)$ denote its exponential-square observable. Suppose that, for every $x\in\mathbb{R}^d$, $S_N(x,t)$ te…