4 problems

For $x\in\mathbb{Z}^2$ and $N\geq 1$, let $\ell_x^N$ be the total number of visits to $x$ by a planar simple random walk started at the origin before exiting $[-N,N]^2$. Define the…

For $n\in\mathbb N$ and $A\in\mathbb N$, define the set of sites visited between one and $A$ times and its associated local time by … Let $\lambda_d(A)$ and $J(A,s)$ denote the qua…

For $d\geq 3$, let $f^{\mathrm{wsaw}}(u)$ be the weakly self-avoiding walk free energy and let $\lambda_d$ be the constant defined from the simple random walk Green function. Weakl…

Let $d\geq 2$, let $\delta$ denote the charge bias, and let $\varepsilon_\delta$ and $\kappa_d$ be the quantities defined in the paper through the small-$\delta$ expansion of the c…