06 Mixing times via super - fast coupling
نویسندگان
چکیده
We provide a coupling proof that the transposition shuffle on a deck of n cards is mixing of rate n log(n) with a moderate constant. This has already been shown by Diaconis and Shahshahani but no natural coupling proof has been demonstrated to date. We also enlarge the methodology of coupling to include intuitive but nonadapted coupling rules, for example, to take in account future events and to prepare for their occurrence.
منابع مشابه
Mixing times via super-fast coupling
We provide a coupling proof that the transposition shuffle on a deck of n cards is mixing of rate Cn log n with a moderate constant, C. This rate was determined by Diaconis and Shahshahani, but the question of a natural probabilistic coupling proof has been missing, and questions of its existence have been raised. The proof, and indeed any proof, requires that we enlarge the methodology of coup...
متن کاملMixing times for the Mean-field Blume-capel Model via Aggregate Path Coupling
We investigate the relationship between the mixing times of the Glauber dynamics of a statistical mechanical system with its thermodynamic equilibrium structure. For this we consider the mean-field Blume-Capel model, one of the simplest statistical mechanical models that exhibits the following intricate phase transition structure: within a two dimensional parameter space there exists a curve at...
متن کاملMarkov chains and mixing times ( part 2 - coupling )
This week’s post continues last week’s discussion of Markov chains and mixing times, and introduces the idea of coupling as a method for estimating mixing times. We remark that some nice notes on the subject of coupling (and others) can be found on Steve Lalley’s web page – of particular relevance for our purposes here are the notes on “Convergence rates of Markov chains”. A more thorough and c...
متن کاملSuppression of Chaos at Slow Variables by Rapidly Mixing Fast Dynamics through Linear Energy-preserving Coupling
Abstract. Chaotic multiscale dynamical systems are common in many areas of science, one of the examples being the interaction of the low-frequency dynamics in the atmosphere with the fast turbulent weather dynamics. One of the key questions about chaotic multiscale systems is how the fast dynamics affects chaos at the slow variables, and, therefore, impacts uncertainty and predictability of the...
متن کاملSuper-GZK Photons from Photon-Axion Mixing
We show that photons with energies above the GZK cutoff can reach us from very distant sources if they mix with light axions in extragalactic magnetic fields. The effect which enables this is the conversion of photons into axions, which are sufficiently weakly coupled to travel large distances unimpeded. These axions then convert back into high energy photons close to the Earth. We show that ph...
متن کامل