We consider the concentration of measure for $n$ i.i.d., two-dimensional random variables under the conditioning that they form a record. Under mild conditions, we show that all random variables tend to concentrate, as $n \rightarrow \infty$, around limiting curves, which are the solutions of an appropriate variational problem. We also show that the same phenomenon occurs, without the records conditioning, for the longest increasing subsequence in the sample.
"Limiting Curves for I.I.D. Records." Ann. Probab. 23 (2) 852 - 878, April, 1995. https://doi.org/10.1214/aop/1176988293