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Here is one way to extend it using a simple union bound to control the deviation of every coordinate at once. You could imagine other ways to extend it, but ensuring that every coordinate is near the …

Schramm's formula on left passage probabilities of $SLE_k$ is stated for $k \in [0,8)$ in theorem 2 here. However, after the statement he remarks that the formula simplifies to $1/2$ for $k = 8$. It s …

Let $\mathbb{D}$ be the unit disc. Let $a,b \in \partial \mathbb{D}$. Let $\gamma$ be a chordal $SLE_{k}$ from $a$ to $b$. For $k \leq 4$, $\gamma$ is a simple curve, and so $\mathbb{D} \setminus \ga …

Consider any $n \times n$ left stochastic matrix $A$, i.e. each column sums up to $1$. We argue that if $H(Ap) \geq H(p)$ for all probability distributions $p$, then $A$ is doubly stochastic. Take $p$ …

Let $G$ be a finite, connected graph. Let $T$ be a uniform spanning tree, and let $e$ be a uniformly random edge not in $T$. When we add $e$ to $T$, we get a subgraph with a unique cycle, $C$. I am in …