21 problems

GUE Tracy–Widom fluctuation conjecture. There are functions $J:\mathbb{R}\to\mathbb{R}$ and $\sigma:\mathbb{R}\to\mathbb{R}$ such that

Let $u{rho,rho}$ be the equilibrium product measure on ${0,1}^{mathbf Z}$ with particle density $rho$ on both sides of the origin, and let $X(t)$ denote the position of a second c…

Fluctuation-type conjecture. For other values of $b3$, the fluctuation is of symplectic-unitary type when $0\leq b3<1$, whereas it is of Gaussian type when $b3>1$.

Let $\rho_-$ and $\rho_+$ be the left and right Bernoulli densities for the TASEP, let $\bar h(y)$ be the hydrodynamic limiting height profile, and let $h_t(j)$ be the height at ti…

Let ${\rm DEP}$ denote the directed exclusion process on the half-line, and let $\nu$ be a nontrivial stationary measure, meaning $\nu\ne\delta_{\mathbf 0},\delta_{\mathbf 1}$. If…

Critical-scale fluctuation conjecture. In dimensions $d>1$, there should similarly be a critical scale at which the nature of the fluctuations changes from Gaussian to non-Gaussian…

Let $d\ge 2$ and consider the facilitated exclusion process (FEP), with critical density $\rho_c$ and stationary measure $\nu_\rho$. Write $\delta=\rho_c-\rho>0$, and let…

The phase boundary of an area-constrained random walk has a maximal local roughness, namely the maximal deviation of the interface from the convex hull of the interface. Its scalin…

The fluid is described by Euler-scale hydrodynamics with macroscopic fields whose characteristic modes may be self-interacting through the dependence of their effective propagation…

Limiting-variance conjecture. One should have

Let the stochastic six vertex model be considered under the scaling regime in which its height function converges to the deterministic telegraph equation, and let the associated ce…

One-third fluctuation exponent conjecture. The fluctuation should be of order

Let $\mathsf{SG}$ be the Sierpinski gasket graph, let $\mathcal{B}_n$ be the graph-metric ball of radius $n$, and let $\mathcal{I}_{|\mathcal{B}_n|}$ be the internal DLA cluster fo…

Let $\mathcal{I}_n$ denote the internal DLA cluster after $n$ particles on $\mathbb{Z}^d$, and measure its fluctuations by the deviation of the cluster from its Euclidean-ball limi…

Let $SG$ be the graphical Sierpinski gasket, let $o$ be its source vertex, and let $B_o(n)$ denote the graph-metric ball of radius $n$ centered at $o$. Write…

Let $d\ge 2$, let $E$ be the set of nearest-neighbor edges of $\mathbb{Z}^d$, and let $(\omega_e)_{e\in E}$ be i.i.d. nonnegative edge weights. For points $x,y\in\mathbb{Z}^d$ at d…

Let $X=X_1\cdots X_n$ and $Y=Y_1\cdots Y_n$ be independent random strings, and let $L_n$ denote their optimal alignment score. In an LPP formulation, $L_n$ is the weight of the hea…

Consider a rarefaction wave with field value $\vec{u}_{\mathsf{v}}$ on the ray $x=\mathsf{v}t$, and let $\vec{\Phi}(x,t)$ denote the integrated fluctuation field. Let…

Let the fluctuation field be the empirical particle-density discrepancy from the profile predicted by the Smoluchowski coagulation-fragmentation-diffusion PDE, normalized by the sq…

For an exclusion process with conductances, let current fluctuations and occupation-time fluctuations denote the fluctuations of the corresponding integrated observables. Folklore…

Cubic-current height fluctuation conjecture. The height at the origin should satisfy