A random graph in $G(n, p)$ model is a graph on $n$ vertices in which for each of the $n\choose{2}$ edges we independently flip a coin, then take the edge with probability $p$ or remove it with $1 - p$.

A random graph in $G(n, m)$ model is a graph on $n$ vertices in which a subset of edges of a fixed size $m$ is chosen at random.

We expect $G(n, p)$ and $G(n, m)$ for $m = p {n\choose{2}}$ to look asymptotically the same, because the number of edges in $G(n, p)$ is highly concentrated around the mean.

The (normalized) Laplacian $L$ on a graph is an operator (matrix) which has entries:

$L(v, v) = 1$

$L(v, w) = - \frac{1}{\sqrt{deg(v)deg(w)}}$

where $v \neq w$ are vertices of the the graph.

We are interested in $\lambda_{2}$, that is, the smallest nonzero eigenvalue of $L$. Suppose we know that, for $p = p(n)$ growing sufficiently fast (papers by Chung et al. show that, for example, $p \geq \frac{\log^2 n}{n}$), $\lambda_{2}$ for a random graph in $G(n, p)$ model is close to 1 with high probability (i. e. approaching 1).

Does it follow immediately that also in $G(n, m)$ we have $\lambda_{2} \to 1$ with high probability? It is known in random graph theory that such implications hold for monotone properties (that is, properties which still hold after adding an edge to the graph), however, the second eigenvalue is not monotone (although it is probably "monotone on average", so I expect the statement above is true for $G(n, m)$).