A random ensemble of cusp forms for the full modular group is introduced. For a weight-k cusp form, restricted to a compact subdomain of the modular surface, the true order of magnitude of its expected supremum is determined to be approximately \sqrt{\log(k)}, in line with the conjectured bounds. In addition, the exponential concentration of the supremum around its median is established. Contrary to the compact case, the global expected supremum, attained around the cusp, grows like k^{1/4}. This talk is based on a forthcoming work, joint with B. Huang, S. Lester and N. Yesha.