Caged black hole thermodynamics: Charge, the extremal limit, and finite size effects

We extend the effective field theory treatment of the thermodynamics of small compactified black holes to the case of charged black holes. The relevant thermodynamic quantities are computed to second order in the parameter \lambda\sim(r_0/L)^(d-3). We discuss how the addition of charge to a caged black hole may delay the phase transition to a black string. In the extremal limit, we construct an exact black hole solution which serves as a check for our perturbative results. Finite size effects are also included through higher order operators in the worldline action. We calculate how the thermodynamic quantities are modified in the presence of these operators, and show they enter beyond order \lambda^2 as in the uncharged case. Finally, we use the exact solution to constrain the Wilson coefficients of the finite size operators in the extremal limit.