15 problems

Let ${P}^0$ be the annealed law of an i.i.d. random walk in random environment on $\mathbb Z^d$, and let $S^{d-1}$ be the set of unit vectors in $\mathbb R^d$. For…

Kalikow's 0-1 conjecture.

Let $(X_n)_{n\geq 0}$ be a random walk in a stationary ergodic environment on $\mathbb{Z}^d$. A deterministic limiting velocity is a vector $v$ such that … for almost every realisa…

RWRE–directed polymer correspondence conjecture. There should be a precise relation between the RWRE and directed-polymer models such that

Directed landscape conjecture. For even larger scales

Critical stochastic heat flow conjecture. In $d=2$, at this scaling, the limiting object should be the 2d critical stochastic heat flow $Z_{s,t}^{\theta}(\mathrm d x,\mathrm d y)$.

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 $N$ 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 ${P}^0$ be the annealed law of an i.i.d. random walk in random environment on $\mathbb Z^d$. For $\ell\in S^{d-1}$, let $A_\ell^0$ be the event that…

Let ${P}^0$ be the annealed law of an i.i.d. random walk in random environment on $\mathbb Z^d$. For $\ell\in S^{d-1}$, let $A_\ell^0$ be the event that…

The transience-condition $(T)$ conjecture. Directional transience along an open set $\mathcal U_{\ell}$ is equivalent to condition $(T)|\ell$.

The strong ballistic conjecture. Any $d$-dimensional random walk in an i.i.d. uniformly elliptic random environment that is directionally transient along an open set of directions…

Distributional-limit conjecture. If $\kappa\in(2,3)$, then

Assume $E[\log\rho_0]<0$, and assume the stronger definition of $\kappa$ together with $E[\rho_0^\kappa\log\rho_0]<\infty$ and a non-lattice distribution for $\log\rho_0$, as descr…

Let $\kappa\in(2,3)$, let $\overline{F}_{n,\omega}$ denote the quenched distribution function appearing in the central limit theorem, let $\Phi$ be the standard normal distribution…