14 problems

Let $\chi_d=\left(\frac{d}{\cdot}\right)$ denote the Kronecker symbol associated with a fundamental discriminant $d$. For $m>0$, define … where $\sideset{}{^*}\sum$ denotes a sum o…

Moment upper-bound conjecture. The moment upper bound can also be established for spatially colored noise.

Counterexample conjecture. The sole assumption $f_{in}\in\bigcap_{s>0}L^1_s$ may not be sufficient to propagate uniformly the $L^1_s$ norm for $s>2$; it may be possible to construc…

Moment-bound conjecture. The same bound holds for all $2\leq p<\infty$.

Let $\beta_N$ be the scaling defined in the paper, and let $chat\beta<\hat\beta_c$. Write $W_N(\beta_N)$ for the normalized partition function. The uniform moment-boundedness conje…

For a matrix tuple $\mathcal X$, consider the determinant of its associated linear pencil evaluated at a tuple $\mathcal U$ of independent Haar unitary matrices. Uniform matrix-coe…

Optimality conjecture. The sufficient condition

Let $W_n$ be the polymer martingale and define … Under the assumption that the weights are bounded, the weak-disorder region is the region where $W_\infty>0$ almost surely, and…

Let $U\subset\mathbb{R}^d$ be a bounded domain satisfying the hypotheses of either Theorem or Theorem, and let $u(t,x)$ be the corresponding solution to the stochastic heat equatio…

Let $\mathbf{X}=SMPR\left(\left\{n^{2}\alpha_{1}\right\},f_{B}(x\mid 1/2,1/2)\right)$. Here $SMPR$ denotes the stationary Markov process specified by its eigenvalue sequence and st…

Let $\psi$ satisfy the growth bounds in \, and let $H(x,v)$ be the function in inequality \. The mesh satisfies $0=t_0\leq t_1\leq\cdots$ with $\delta_k=t_k-t_{k-1}<\delta_0$,…

Consider sufficiently spread-out oriented percolation in dimension $d>4$, with spread parameter $L$ sufficiently large, and let $\mathcal {T}_n$ denote the set of vertices connecte…

Consider the hybrid linear system with jumps … where $W_t$ is a Wiener process, $N_t$ is a counting process, and $L_t$ satisfies a moment condition of order $Q$. Moment-boundedness…

Moment-bound conjecture. The estimate