17 problems

Let $u_*(d,T)$ denote the critical intensity for percolation in finitary random interlacements on $\mathbb{Z}^d$, and let $f_d(T)$ be given by … Suppose that, for sufficiently larg…

Let $h_*$ and $u_*$ denote, respectively, the critical parameters for level-set percolation of the discrete Gaussian free field and for percolation of the vacant set of random inte…

Low-intensity Euclidean geometry conjecture. As $u\to 0$, the geometry of $\mathcal{I}^u$ converges to Euclidean geometry; equivalently, $D^u$, after the appropriate scaling, conve…

Sharp-constant conjecture. In each dimension, the lower- and upper-bound constants can be chosen arbitrarily close to one another.

Inverse-order conjecture. The critical intensity satisfies

For each $d\ge 3$, let $T_c(u)$ be the critical value from the preceding conjecture. Asymptotic critical-value conjecture. There is a constant $c_d$ such that … The conjecture is b…

Let $d\ge 3$ and $u>0$, and let $\mathcal{FI}^{u,T}_d$ be finitary random interlacement with parameters $u$ and $T$. Unique-infinite-cluster conjecture. There is a critical value…

Let $d\ge 3$ and $u>0$, and let $\mathcal{FI}_d^{u,T}$ denote finitary random interlacement on the lattice with parameters $u$ and $T$. A critical value $T_c(u,d)$ is a threshold f…

Nonpercolation conjecture. There are no infinite components in $\mathcal{K}$.

Fix $u>0$. Let $d_{\mathcal{FI}^{u,T}}(\cdot,\cdot)$ and $d_{\mathcal{I}^{u}}(\cdot,\cdot)$ denote the internal graph distances in finitary random interlacements (FRI) and random i…

Consider the continuum model formed by a Poisson cloud of Brownian motion paths or Wiener sausages, truncated at time $t$, with intensity parameter $\lambda>0$. Let continuous rand…

Let $d\geq 4$, let $\beta>1$ be the bias parameter, and let $u>0$ be the intensity parameter of the random interlacements. Consider the biased random walk on the interlacement clus…

For $A\subset \mathbb Z^d$, let $\partial A=\{x\in A:\text{there exists }y\notin A\text{ such that }x\sim y\}$ be its internal boundary. For simple random walk and finite…

Let the vacant set be the set of vertices of the discrete $d$-dimensional torus not visited by a random walk run up to time $uN^d$, where $N$ is the sidelength and $u$ is the time…

Let $\mathbb{T}_N=({\mathbb{Z}} / N {\mathbb{Z}})^d$ with $d\geq 3$, and let a simple symmetric nearest-neighbor random walk started uniformly on $\mathbb{T}_N$ visit the set…

Let $d\geq 3$. For $u\geq 0$, let $\mathcal V^u$ denote the vacant set of random interlacements, and let $\mathcal V^{u,\varepsilon}$ denote its $\varepsilon$-disordered version. D…

Let ${\cal V}^u$ be the vacant set at level $u\geq 0$ of random interlacements on the relevant lattice, and let $u_*$ denote its critical percolation parameter. Consider the fragme…