Arguably mathematics is getting harder, although some people claim that also in the old times parts of it were hard and known only to a few experts before major simplifications had changed matters. Let me report here about two recent remarkable simplifications of major theorems. I am thankful to Nati Linial who told me about the first and to Itai Benjamini and Gady Kozma who told me about the second. Enjoy!

## Random regular graphs are nearly Ramanujan: Charles Bordenave gives a new proof of Friedman’s second eigenvalue Theorem and its extension to random lifts

**Abstract:** It was conjectured by Alon and proved by Friedman that a random -regular graph has nearly the largest possible spectral gap, more precisely, the largest absolute value of the non-trivial eigenvalues of its adjacency matrix is at most with probability tending to one as the size of the graph tends to infinity. We give a new proof of this statement. We also study related questions on random *n*-lifts of graphs and improve a recent result by Friedman and Kohler.

**A simple proof for the theorem of Aizenman and Barsky and of Menshikov. Hugo Duminil-Copin and Vincent Tassion give ** a new proof of the sharpness of the phase transition for Bernoulli percolation on

**Abstract:** We provide a new proof of the sharpness of the phase transition for nearest-neighbour Bernoulli percolation. More precisely, we show that

– for , the probability that the origin is connected by an open path to distance $n$ decays exponentially fast in $n$.

– for , the probability that the origin belongs to an infinite cluster satisfies the mean-field lower bound .

This note presents the argument of this paper by the same authors, which is valid for long-range Bernoulli percolation (and for the Ising model) on arbitrary transitive graphs in the simpler framework of nearest-neighbour Bernoulli percolation on .