4 problems

Phase-transition conjecture. There exists a non-trivial critical value $u_c=u_c(d)\in(0,\infty)$, independent of $N$, such that $\mathcal{V}_N^u$ undergoes an abrupt phase transiti…

Consider the VSRW model on random graphs with a general prescribed degree sequence in the critical regime, and assume the degree distribution has a heavy tail with exponent…

Let $\boldsymbol{d}=\boldsymbol{d}^{(n)}=(d_1,\ldots,d_n)$ be a degree sequence, and let $D_n$ denote the degree of a uniformly chosen vertex in $[n]$. Assume that $D_n$ converges…

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…