Gauge/gravity duality model for gauge theory thermodynamics

We study a gauge/gravity model for the thermodynamics of a gauge theory with one running coupling. The gravity side contains an ansatz for the metric and a scalar field; on the field theory side one starts by giving an ansatz for the beta function describing the scale dependence of the coupling. The model is based on relating the scale to the extra dimensional coordinate and the beta function to the gravity fields, thereby also determining the scalar field potential. We study three different forms of beta functions of increasing complexity and give semianalytic solutions describing first order or continuous transitions.