56 problems

A generalized Hamilton–Jacobi polymer is a directed polymer model with a random potential whose path weights are governed by a generalized Hamilton–Jacobi structure. In this settin…

Successive-merging conjecture. In dimension $d=3$, the curves in Fig. 3 never merge, whereas for $d\geq 4$ the curves merge successively.

Let $T$ denote the first-passage percolation passage time, with expectation $\mathbb{E}[T]$ and variance $\operatorname{var}(T)$. The two-thirds variance exponent conjecture. The v…

High-dimensional non-intermittency conjecture. There is no intermittency for sufficiently large diffusion constant $\kappa$ in dimensions $d\ge 4$.

Let $p\in(p_c,1)$, where $p_c=\frac{1}{2}$ is the critical probability for bond percolation on $\mathbb{Z}^2$. For the biased random walk on the infinite percolation cluster, let…

Consider an irreducible $n$-component system in which the conductivity $sigma$ takes the distinct values $\sigma_1,\ldots,\sigma_n$ with positive probabilities $p_1,\ldots,p_n$ sat…

Let $a_{1,2}(t,\mathbf{x},\mathbf{k})$ denote the Wigner-distribution components associated with the two wave modes in the random hyperbolic system. Ensemble averaging means averag…

Graph distance is the intrinsic metric induced by connectivity in a random medium. In several models of random media, let the graph distance measure the length of shortest paths be…

Consider the mirrors model on the lattice $\mathbb{Z}^d$, with a mirror placed independently at each site with probability $p$, and let a particle move along lattice edges, choosin…

KPZ fluctuation conjecture. At the critical scaling, one-dimensional diffusions in random media should exhibit fluctuations governed by the KPZ equation.

Let $\theta(t,x)$ denote the random field solving the stochastic advection-diffusion equation under consideration, and consider its behavior in the diffusive scaling regime. Edward…

Let $\gamma$ be the model parameter, let $\Delta_\gamma(H_2)$ denote the relevant boundary curve, and let $\alpha(\gamma)$ be the exponent defined by the preceding model equations.…

High-frequency waves propagating through randomly varying media are modeled by a wavefield whose distribution is defined in the long-distance limit. Gaussian conjecture. High-frequ…

Let a wave field propagate over a long distance through a random medium, and let its scintillation index be the ratio of the variance of the intensity to the square of the average…

KPZ tail-fluctuation conjecture. The fluctuations of the random field $P^\omega(r,y)$ should be described by the KPZ equation at some point within the tail of that probability meas…

Polyakov's conjecture. This $\beta$-dependent quenched disorder should destroy the power-law decay of the induced $XY$ model for every $\beta>0$.

Let $N$ denote the scaling parameter, and consider the fluctuations of quenched densities and related quantities in stochastic flows arising from random walks in random environment…

Let $x\in\mathbb{Z}^2$ be a root and let $\mathcal V^{\omega}$ denote the set of jump directions of the Busemann process. For $\xi\in\mathcal V^{\omega}$, consider the $\xi$-direct…

KPZ shape-function conjecture. The shape functions are strictly convex and differentiable; equivalently, they have no flat edges and no corners.

Depinning roughness conjecture. Based on numerical simulations, the exponent is

Consider the macroscopic wave-equation approximation for a lattice with i.i.d. random masses, and denote by a.e.d. and a.e.v. the absolute error measures used in the source. Three-…

Let $c3$ be the small positive parameter appearing in the error estimates, and let $c7$ be the cut-off corrector whose equation is denoted by … . This is a tentative expectation ra…

Superdiffusivity conjecture. Under strong disorder, the polymer measure should belong to the KPZ universality class, and the paths should have typical fluctuations of order

Favorite region conjecture. Under strong disorder, the midpoint, or any other point, distribution of a point-to-point directed polymer should be asymptotically localized in a regio…

$L^1$ convergence conjecture. There exists a constant $M\in(-\infty,+\infty)$ such that