We study Mallows random permutations conditioned to avoid a given pattern α of length 3, for which we find limit shapes in the space of permutons when the bias parameter is of the form e^(β/n). Along the way, we provide parametrizations for α-avoiding permutons, and establish large deviation principles for uniform α-avoiding permutations.

This paper introduces a unified framework for the detection of a single source with a sensor array in the context where the noise variance and the channel between the source and the sensors are unknown at the receiver. The Generalized Maximum Likelihood Test is studied and yields the analysis of the ratio between the maximum eigenvalue of the sampled covariance matrix and its normalized trace. Using recent results from random matrix theory, a practical way to evaluate the threshold and thep-value of the test is provided in the asymptotic regime where the numberKof sensors and the numberNof observations per sensor are large but have the same order of magnitude. The theoretical performance of the test is then analyzed in terms of Receiver Operating Characteristic (ROC) curve. It is, in particular, proved that both Type I and Type II error probabilities converge to zero exponentially as the dimensions increase at the same rate, and closed-form expressions are provided for the error exponents. These theoretical results rely on a precise description of the large deviations of the largest eigenvalue of spiked random matrix models, and establish that the presented test asymptotically outperforms the popular test based on the condition number of the sampled covariance matrix.