25 problems

Let $W$ be the normalized sum considered in the paper, let $\kappa_j(W)$ denote its $j$-th cumulant, and let $f\in\mathcal{C}^{k,\omega}(\mathbb{R})$, where $k\in\mathbb{N}_+$ and…

Let $\lambda^1,\dots,\lambda^r$ be partitions. For $I\subseteq [r]$, let $\lambda^I$ denote the combined partition used in the Macdonald polynomial $J_{\lambda^I}^{(q,t)}$, and let…

Let bi-non-crossing partitions be the combinatorial objects introduced by Mastnak and Nica, and let the associated $(l,r)$-cumulants be the cumulants indexed by these partitions. M…

Let $C_{k,m,n}$ be the minimum cost of a $k$-matching in the random bipartite matching model, and let its cumulants $\kappa_p$ be defined by … where…

Let $\eta>1$ be algebraic but not a Perron number, and consider a lacunary sequence of the form … Let $S_n$ denote the associated lacunary trigonometric sum, and let…

Conjectured coefficients.

For general $q$, let $f_1$ be the primitive arising from the first term in the integrand, and let $\Delta_{2q}(k)$ denote the corresponding function whose $(2q-1)$st derivative at…

Let $J=J(0,T)$ be the current, and consider equal reservoir densities $\rho_1=\rho_2=\rho=1/2$. Polynomial-cumulant conjecture. The cumulants of $J$ are simply polynomials in the a…

Let $q$ and $k$ be large, let $V_1$ and $V_2$ be the model-dependent quantities appearing in the limiting cumulants, and set … Let $\upsilon(s)$ be the function defined through the…

Higher-order cumulant oscillation conjecture. Higher-order cumulants, which are rational functions of the utilization, also exhibit oscillations.

Let $\xi_n$ be the random variable with … where $B_n=T_n(1)$ is the $n$-th Bell number, and let $\kappa^{(m)}(\xi_n)$ denote its $m$-th cumulant. Cumulant asymptotics conjecture. F…

Let $T_r$ denote the correction to the $r$th cumulant of $X^{(n)}_3$ arising from dependence among the pair indicators. Cumulant-correction conjecture. … The surrounding argument e…

General smooth-potential conjecture. In the large-$\mu$ limit,

Let $X_1,X_2,\dots,X_n\in\mathcal{A}_{sa}$ be a free family of random variables, and let $P$ be a selfadjoint polynomial of degree $2$ in noncommuting variables. Say that $P$ exhib…

Let $F_\infty$ be the target random variable in the second Wiener chaos of the form referenced in the source, and let $F=I_q(f)$ be a chaotic random variable in the $q$th Wiener ch…

Consider the cumulant expansion of the moment-generating function for the one-dimensional KPZ equation, with $\gamma_n(s)$ denoting the contribution from the $n$-th cumulant. In th…

Conditional-expectation conjecture. The Betti numbers can be obtained as cumulants with respect to a suitable conditional expectation.

Cumulant-coefficient positivity conjecture. For partitions $cpi_1,\dots,cpi_\ell$, the polynomial $(-1)^{\ell-1}d^\sigma_{\pi_1,\dots,\pi_\ell}(cdelta)$ has non-negative integer co…

Kerov–Lassalle cumulant-positivity conjecture. For all integers $k_1,\dots,k_\ell\geq 1$, the polynomial which expresses this signed cumulant in terms of…

Time-correlation evolution conjecture. The equation

Let $\pazocal{R}_3(\mathbf{k},t)$ denote the remainder arising from the third-cumulant correction in the perturbation expansion, and let $\tau\lambda^{-2}$ be kinetic time. Third-r…

Let $W_t^{\lambda}(\mathbf{k})$ be the Fourier transform of the second cumulant, and let $\tau\geq 0$ denote kinetic time. The kinetic scaling is $t=\tau\lambda^{-2}$. Kinetic-limi…

Let $X$ be a multiple Wiener--Itô integral of order $p\geq 2$, and let $\kappa_6(X)$ denote its sixth cumulant. Sixth-cumulant positivity conjecture. For every such $X$, one has ……

Cumulant conjecture. The right-hand side of this relation must be related to the $p$th cumulant of the $p$th moment of the mean spectral distribution of $M$, and this relation shou…

Let $\b7$?