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Fix some positive integers $p,n,k$. Let $w$ be chosen uniformly at random from $[k]^n$ (the set of $n$ length words/sequences where each entry is in $\{1,\ldots,k\}$). Let $A_i$ be the event that $w_ …

Consider a random walk $S_n=\sum_{i=1}^n X_i$ where $P(X_i=+1)=P(X_i=-1)=1/2$ with $n$ large. By Chernoff's bound we know that, for example, $\sum_{i=1}^{n/2} X_i=O(\sqrt{n})$ with high probability. …

I believe the following coupling argument shows that (in particular) if we specify that the random walk ends at 0 then halfway through the walk the probability that we're within distance $\lambda$ of …

In their paper, Erdos and Renyi consider a random graph with a fixed number of edges, as opposed to the more modern approach of adding each edge independently with probability $p$. From what I unders …

The most satisfying solution I've seen so far are these notes by Sabastian Roch (see section 2.2.3, and in particular claim 2.25). He first argues that the only components besides the giant one are i …