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Register a new userMetastability and small eigenvalues in Markov chains
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A. Bovier
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In this letter we announce rigorous results that elucidate the relation between metastable states and low-lying eigenvalues in Markov chains in a much more general setting and with considerable greater precision as was so far available. This includes a sharp uncertainty principle relating all low-lying eigenvalues to mean times of metastable transitions, a relation between the support of eigenfunctions and the attractor of a metastable state, and sharp estimates on the convergence of probability distribution of the metastable transition times to the exponential distribution.
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We study a large class of reversible Markov chains with discrete state space and transition matrix $P_N$. We define the notion of a set of {it metastable points} as a subset of the state space $G_N$ such that (i) this set is reached from any point $xin G_N$ without return to x with probability at least $b_N$, while (ii) for any two point x,y in the metastable set, the probability $T^{-1}_{x,y}$ to reach y from x without return to x is smaller than $a_N^{-1}ll b_N$. Under some additional non-degeneracy assumption, we show that in such a situation: item{(i)} To each metastable point corresponds a metastable state, whose mean exit time can be computed precisely. item{(ii)} To each metastable point corresponds one simple eigenvalue of $1-P_N$ which is essentially equal to the inverse mean exit time from this state. The corresponding eigenfunctions are close to the indicator function of the support of the metastable state. Moreover, these results imply very sharp uniform control of the deviation of the probability distribution of metastable exit times from the exponential distribution.
We study a class of Markov chains that describe reversible stochastic dynamics of a large class of disordered mean field models at low temperatures. Our main purpose is to give a precise relation between the metastable time scales in the problem to the properties of the rate functions of the corresponding Gibbs measures. We derive the analog of the Wentzell-Freidlin theory in this case, showing that any transition can be decomposed, with probability exponentially close to one, into a deterministic sequence of ``admissible transitions. For these admissible transitions we give upper and lower bounds on the expected transition times that differ only by a constant. The distribution rescaled transition times are shown to converge to the exponential distribution. We exemplify our results in the context of the random field Curie-Weiss model.
Quasiperiodic systems are aperiodic but deterministic, so their critical behavior differs from that of clean systems as well as disordered ones. Quasiperiodic criticality was previously understood only in the special limit where the couplings follow discrete quasiperiodic sequences. Here we consider generic quasiperiodic modulations; we find, remarkably, that for a wide class of spin chains, generic quasiperiodic modulations flow to discrete sequences under a real-space renormalization group transformation. These discrete sequences are therefore fixed points of a emph{functional} renormalization group. This observation allows for an asymptotically exact treatment of the critical points. We use this approach to analyze the quasiperiodic Heisenberg, Ising, and Potts spin chains, as well as a phenomenological model for the quasiperiodic many-body localization transition.
Using strong-disorder renormalization group, numerical exact diagonalization, and quantum Monte Carlo methods, we revisit the random antiferromagnetic XXZ spin-1/2 chain focusing on the long-length and ground-state behavior of the average time-independent spin-spin correlation function C(l)=upsilon l^{-eta}. In addition to the well-known universal (disorder-independent) power-law exponent eta=2, we find interesting universal features displayed by the prefactor upsilon=upsilon_o/3, if l is odd, and upsilon=upsilon_e/3, otherwise. Although upsilon_o and upsilon_e are nonuniversal (disorder dependent) and distinct in magnitude, the combination upsilon_o + upsilon_e = -1/4 is universal if C is computed along the symmetric (longitudinal) axis. The origin of the nonuniversalities of the prefactors is discussed in the renormalization-group framework where a solvable toy model is considered. Moreover, we relate the average correlation function with the average entanglement entropy, whose amplitude has been recently shown to be universal. The nonuniversalities of the prefactors are shown to contribute only to surface terms of the entropy. Finally, we discuss the experimental relevance of our results by computing the structure factor whose scaling properties, interestingly, depend on the correlation prefactors.
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