Has the technique of "sprinkling" been used in studying random matrices?

Question

In 1982, while studying the component sizes of random subgraphs of a hypercube, Ajtai, Komlós, and Szemerédi introduced a technique that came to be known as sprinkling. In this technique, the edges of the random graph are exposed in rounds. To explain it, suppose that each edge $e$ is independently assigned a random Uniform$[0,1]$ variable $U_e$. Eventually, all edges with $U_e \leq p$ will be included in the graph. In the first round, however, for some subset of the edges, we only check whether $U_e \leq p-\epsilon$. In the second round, for the remaining edges for which we know $U_e > p-\epsilon$, we check whether $U_e \leq p$. (The last few edges are the ones being "sprinkled" on at the end.) The idea is that this additional, last-minute randomization can be used to ensure (or at least make it very likely) that some desirable graph property holds. A similar technique has also been used by percolation theorists.

Has the technique of sprinkling been used in the study of random Bernoulli matrices? Can you give me references?

Answer 1

The continuous comparison method of Knowles and Yin,

Knowles, Antti; Yin, Jun, Anisotropic local laws for random matrices, Probab. Theory Relat. Fields 169, No. 1-2, 257-352 (2017). ZBL1382.15051.

follows a strategy like this. To continuously deform one random matrix (e.g., a Wigner matrix) $X^0$ to another $X^1$, assign independent uniform random variables $U_{ij} \in [0,1]$ to each matrix entry, and for each time $0 \leq \theta \leq 1$, let $X^\theta$ be the matrix whose $ij^{th}$ entry is that of $X^0$ if $U_{ij} > \theta$ and that of $X^1$ otherwise. This gives a family of random matrices that continuously interpolate between $X^0$ and $X^1$ in the vague topology, and one can start comparing matrix statistics of $X^1$ to that of $X^0$ by differentiating in $\theta$ and using the fundamental theorem of calculus. This is similar to the Lindeberg exchange method which is also commonly used in this field, in which one replaces the entries of $X^0$ with that of $X^1$ one at a time (or two at a time, if one wants to preserve the Hermitian property), but there are some additional averaging effects (analogous to the randomising effects of the "sprinkling" that you mention) that can be exploited sometimes in the continuous version of this strategy.

See for instance these slides of Jun Yin for a description of the method.

Another similar method that comes to mind is the "switching" methods used to study random regular graph (or digraph) models, for instance in

Cook, Nicholas A., Discrepancy properties for random regular digraphs, Random Struct. Algorithms 50, No. 1, 23-58 (2017). ZBL1352.05164.

where the invariance of random regular (di)-graphs with respect to permutations (or to "switching" edges within a quadruplet of vertices) is used to sprinkle in some useful independent randomness into the ensemble for various purposes.