In this talk I aim to discuss asymptotically almost sure (aka a.a.s.) expansion lower bounds of the uniform ensemble of d-regular graphs and show that random π-lifts may be used to obtain some improvements. In this regard, after introducing the model, I briefly go through some techniques already used in this area of research and based on a recent joint contribution with MH. Shojaedin, I will introduce a general reduction method that provides a.a.s. lower bounds when a.a.s. upper bounds are known, which is based on the analysis of a contiguous ensemble constructed through random π-lifts. This model gives rise to a dual approximation as a conditional optimization problem that can be handled using Bernstein and normal approximation schemes. In particular, I will report some consequences of this approach as improvements of existing a.a.s. lower bounds for the case of small degrees and the case of asymptotically large degrees. Subject to my time limitations I may also talk about the iterated π-lift model and its spectral properties as well as some connections of the subject to statistical mechanics and physics.

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