In "The mixing time of the giant component of a random graph" by the aforementioned authors, in the last proof on page 19 it says something along the lines of

"It is well known and easy to verify that, since $m=O(n)$ , asymptotically almost surely the maximum degree occurring in $\mathcal{C}_3(n,m)$ is $o(n^{0.02})$"

where $\mathcal{C}(n,m)$ denotes a random graph with $n$ vertices, $m$ edges, uniformly sampled among such graphs, and $\mathcal{C}_3(n,m)$ denotes the variable conditioned on minimum degree 3 (which shouldn't influence the above statement).

I am wondering where the $n^{0.02}$ comes from, and how this is "well known". Can someone provide an original source for this statement or this kind of statement?