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Register a new userInfinite DLR Measures and Volume-Type Phase Transitions on Countable Markov Shifts
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We consider the natural definition of DLR measure in the setting of $sigma$-finite measures on countable Markov shifts. We prove that the set of DLR measures contains the set of conformal measures associated with Walters potentials. In the BIP case, or when the potential normalizes the Ruelles operator, we prove that the notions of DLR and conformal coincide. On the standard renewal shift, we study the problem of describing the cases when the set of the eigenmeasures jumps from finite to infinite measures when we consider high and low temperatures, respectively. For this particular shift, we prove that there always exist finite DLR measures, and we have an expression to the critical temperature for this volume-type phase transition, which occurs only for potentials with the infinite first variation.
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For a finitely irreducible countable Markov shift and a potential with summable variations, we provide a condition on the associated pressure function which ensures that Bowens Gibbs state, the equilibrium state, and the minimizer of the level-2 large deviations rate function are all unique and they coincide. From this, we deduce that all periodic points weighted with the potential equidistribute with respect to the Gibbs-equilibrium state as the periods tend to infinity. Applications are given to the Gauss map, and the Bowen-Series map associated with a finitely generated free Fuchsian group with parabolic elements.
In recent years, dynamical quantum phase transitions (DQPTs) have emerged as a useful theoretical concept to characterize nonequilibrium states of quantum matter. DQPTs are marked by singular behavior in an textit{effective free energy} $lambda(t)$, which, however, is a global measure, making its experimental or theoretical detection challenging in general. We introduce two local measures for the detection of DQPTs with the advantage of requiring fewer resources than the full effective free energy. The first, called the textit{real-local} effective free energy $lambda_M(t)$, is defined in real space and is therefore suitable for systems where locally resolved measurements are directly accessible such as in quantum-simulator experiments involving Rydberg atoms or trapped ions. We test $lambda_M(t)$ in Ising chains with nearest-neighbor and power-law interactions, and find that this measure allows extraction of the universal critical behavior of DQPTs. The second measure we introduce is the textit{momentum-local} effective free energy $lambda_k(t)$, which is targeted at systems where momentum-resolved quantities are more naturally accessible, such as through time-of-flight measurements in ultracold atoms. We benchmark $lambda_k(t)$ for the Kitaev chain, a paradigmatic system for topological quantum matter, in the presence of weak interactions. Our introduced local measures for effective free energies can further facilitate the detection of DQPTs in modern quantum-simulator experiments.
We show that under rather general assumptions on the form of the entropy function, the energy balance equation for a system in thermodynamic equilibrium is equivalent to a set of nonlinear equations of hydrodynamic type. This set of equations is integrable via the method of the characteristics and it provides the equation of state for the gas. The shock wave catastrophe set identifies the phase transition. A family of explicitly solvable models of non-hydrodynamic type such as the classical plasma and the ideal Bose gas are also discussed.
For a large class of irreducible shift spaces $XsubsettA^{Z^d}$, with $tA$ a finite alphabet, and for absolutely summable potentials $Phi$, we prove that equilibrium measures for $Phi$ are weak Gibbs measures. In particular, for $d=1$, the result holds for irreducible sofic shifts.
In this note we give asymptotic estimates for the volume growth associated to suitable infinite graphs. Our main application is to give an asymptotic estimate for volume growth associated to translation surfaces.
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