We study the thermodynamic formalism of locally compact Markov shifts with transient potential functions. In particular, we show that the Ruelle operator admits positive continuous eigenfunctions and positive Radon eigenmeasures in forms of Martin kernels. These eigenmeasures can be characterized in terms of the direction of escape to infinity of their orbits, when viewed inside a suitable Martin-like compactification of the underlying shift space. We relate these results to first-order phase transitions in one-dimensional lattice gas models with infinite set of states. This work complements earlier works by Sarig [32,33] who focused on the recurrent scenario. * ofer.shwartz@weizmann.ac.il 2. Representation of eigenvectors: We extend Martin's representation theorem [23] to the context of Ruelle operator. Specifically, we construct a compactification X + of X + with boundary M = X + \ X + and construct a kernel K(f, ω|λ) (f ∈ C + c (X + ), ω ∈ X + ) s.t. every Radon measure µ with L * φ µ = λµ has the form µ(·) = M K(·, ω|λ)dν(ω) 1.1 Topological Markov shifts, Ruelle operator and transience Let S be an infinite countable set of states and let A ∈ {0, 1} S×S be a transition matrix over S. For a subset A ⊆ Z and a vector x ∈ S A , we denote by (x) i the i-th coordinate of x. The (positive) one-sided topological Markov shift (TMS) is the spacewith the transformation T : X + → X + , (T (x)) i = (x) i+1 and the metric d(x, y) = 2 − inf{i≥0:(x)i =(y)i} .(1)

The largest eigenvalue of a Wishart matrix, known as Roy's largest root (RLR), plays an important role in a variety of applications. Most works to date derived approximations to its distribution under various asymptotic regimes, such as degrees of freedom, dimension, or both tending to infinity. However, several applications involve finite and relative small parameters, for which the above approximations may be inaccurate. Recently, via a small noise perturbation approach with fixed dimension and degrees of freedom, Johnstone and Nadler derived simple yet accurate stochastic approximations to the distribution of Roy's largest root in the real valued case, under a rank-one alternative. In this paper, we extend their results to the complex valued case. Furthermore, we analyze the behavior of the leading eigenvector by developing new stochastic approximations. Specifically, we derive simple stochastic approximations to the distribution of the largest eigenvalue under five common complex single-matrix and double-matrix scenarios. We then apply these results to investigate several problems in signal detection and communications. In particular, we analyze the performance of RLR detector in cognitive radio spectrum sensing and constant-modulus signal detection in the high signal-to-noise ratio (SNR) regime. Moreover, we address the problem of determining the optimal transmit-receive antenna configuration (here optimality is in the sense of outage minimization) for rank-one multiple-input and multiple-output Rician-Fading channels at high SNR.

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In this paper we study the conformal measures of a normal subgroup of a cocompact Fuchsian group. In particular, we relate the extremal conformal measures to the eigenmeasures of a suitable Ruelle operator. Using Ancona’s theorem, adapted to the Ruelle operator setting, we show that if the group of deck transformations G is hyperbolic then the extremal conformal measures and the hyperbolic boundary of G coincide. We then interpret these results in terms of the asymptotic behavior of cutting sequences of geodesics on a regular cover of a compact hyperbolic surface.

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