Matthew Tointon

Isoperimetric inequalities in induced subgraphs

Isoperimetric inequalities in graphs (i.e. lower bounds on the size of a set’s boundary in terms of the size of the set itself) can be used to establish various probabilistic properties - two examples particularly relevant to this talk are transience of the random walk, and non-triviality of the percolation phase transition. Recent advances in the structure theory of transitive graphs give essentially optimal and highly useful isoperimetric inequalities in graphs with polynomial growth and their finite analogues. However, in applications it is often the case that one needs an isoperimetric inequality in certain induced subgraphs, and here no useful general inequality seems to be known. In this talk I will describe some recent and forthcoming applications where such an inequality would have been helpful, present some partial results that ended up being good enough for these applications, and briefly speculate on what a general result might look like.