Rigorous Upper Bound for the Discrete Bak–Sneppen Model

Fix some $$p\in [0,1]$$ p ∈ [ 0 , 1 ] and a positive integer n. The discrete Bak–Sneppen model is a Markov chain on the space of zero-one sequences of length n with periodic boundary conditions. At each moment of time a minimum element (typically, zero) is chosen with equal probability, and it is then replaced alongside both its neighbours by independent Bernoulli(p) random variables. Let $$\nu ^{(n)}(p)$$ ν ( n ) ( p ) be the probability that an element of this sequence equals one under the stationary distribution of this Markov chain. It was shown in Barbay and Kenyon (in Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms (Washington, DC, 2001), pp. 928–933, SIAM, Philadelphia, PA, 2001) that $$\nu ^{(n)}(p)\rightarrow 1$$ ν ( n ) ( p ) → 1 as $$n\rightarrow \infty $$ n → ∞ when $$p>0.54\dots $$ p > 0.54 ⋯ ; the proof there is, alas, not rigorous. The complimentary fact that $$\displaystyle \limsup _{n\rightarrow \infty } \nu ^{(n)}(p)< 1$$ lim sup n → ∞ ν ( n ) ( p ) < 1 for $$p\in (0,p')$$ p ∈ ( 0 , p ′ ) for some $$p'>0$$ p ′ > 0 is much harder; this was eventually shown in Meester and Znamenski (J Stat Phys 109:987–1004, 2002). The purpose of this note is to provide a rigorous proof of the result from Barbay and Kenyon (in Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms (Washington, DC, 2001), pp. 928–933, SIAM, Philadelphia, PA, 2001), as well as to improve it, by showing that $$\nu ^{(n)}(p)\rightarrow 1$$ ν ( n ) ( p ) → 1 when $$p>0.45$$ p > 0.45 . (Our method, in fact, shows that with some finer tuning the same is true for $$p>0.419533$$ p > 0.419533 .)

On the empirical spectral distribution for certain models related to sample covariance matrices with different correlations

Given [Formula: see text], we study two classes of large random matrices of the form [Formula: see text] where for every [Formula: see text], [Formula: see text] are iid copies of a random variable [Formula: see text], [Formula: see text], [Formula: see text] are two (not necessarily independent) sets of independent random vectors having different covariance matrices and generating well concentrated bilinear forms. We consider two main asymptotic regimes as [Formula: see text]: a standard one, where [Formula: see text], and a slightly modified one, where [Formula: see text] and [Formula: see text] while [Formula: see text] for some [Formula: see text]. Assuming that vectors [Formula: see text] and [Formula: see text] are normalized and isotropic “in average”, we prove the convergence in probability of the empirical spectral distributions of [Formula: see text] and [Formula: see text] to a version of the Marchenko–Pastur law and the so-called effective medium spectral distribution, correspondingly. In particular, choosing normalized Rademacher random variables as [Formula: see text], in the modified regime one can get a shifted semicircle and semicircle laws. We also apply our results to the certain classes of matrices having block structures, which were studied in [G. M. Cicuta, J. Krausser, R. Milkus and A. Zaccone, Unifying model for random matrix theory in arbitrary space dimensions, Phys. Rev. E 97(3) (2018) 032113, MR3789138; M. Pernici and G. M. Cicuta, Proof of a conjecture on the infinite dimension limit of a unifying model for random matrix theory, J. Stat. Phys. 175(2) (2019) 384–401, MR3968860].

A mechanical model of Brownian motion for one massive particle including low energy light particles in dimension d ≥ 3

We provide a connection between Brownian motion and a classical Newton mechanical system in dimension d ≥ 3 {d\geq 3} . This paper is an extension of [S. Liang, A mechanical model of Brownian motion for one massive particle including slow light particles, J. Stat. Phys. 170 2018, 2, 286–350]. Precisely, we consider a system of one massive particle interacting with an ideal gas, evolved according to non-random Newton mechanical principles, via interaction potentials, without any assumption requiring that the initial energies of the environmental particles should be restricted to be “high enough”. We prove that, as in the high-dimensional case, the position/velocity process of the massive particle converges to a diffusion process when the mass of the environmental particles converges to 0, while the density and the velocities of them go to infinity.

How Many Inflections are There in the Lyapunov Spectrum?

Iommi and Kiwi (J Stat Phys 135:535–546, 2009) showed that the Lyapunov spectrum of an expanding map need not be concave, and posed various problems concerning the possible number of inflection points. In this paper we answer a conjecture in Iommi and Kiwi (2009) by proving that the Lyapunov spectrum of a two branch piecewise linear map has at most two points of inflection. We then answer a question in Iommi and Kiwi (2009) by proving that there exist finite branch piecewise linear maps whose Lyapunov spectra have arbitrarily many points of inflection. This approach is used to exhibit a countable branch piecewise linear map whose Lyapunov spectrum has infinitely many points of inflection.

Про конструктивні описи мір Гіббса для моделі Поттса на дереві Келі

УДК 517.9 Розглядається модель Поттса на деревi Келi. Доведено iснування мiр Гiббса, побудованих аналогiчним методом iз [H. Akin, U. A. Rozikov, S. Temir, <em>A new set of limiting Gibbs measures for the Ising model on a Cayley tree</em>, J. Stat. Phys., <strong>142</strong>, № 2, 314 – 321 (2011)] i -трансляцiйно-iнварiантних мiр Гiббса для моделi Поттса на деревi Келi. Обчислено вiльнi енергiї цих мiр Гiббса.

On the Boundary Layer Equations with Phase Transition in the Kinetic Theory of Gases

Consider the steady Boltzmann equation with slab symmetry for a monatomic, hard sphere gas in a half space. At the boundary of the half space, it is assumed that the gas is in contact with its condensed phase. The present paper discusses the existence and uniqueness of a uniformly decaying boundary layer type solution of the Boltzmann equation in this situation, in the vicinity of the Maxwellian equilibrium with zero bulk velocity, with the same temperature as that of the condensed phase, and whose pressure is the saturating vapor pressure at the temperature of the interface. This problem has been extensively studied, first by Sone, Aoki and their collaborators, by means of careful numerical simulations. See section 2 of (Bardos et al. in J Stat Phys 124:275–300, 2006) for a very detailed presentation of these works. More recently, Liu and Yu (Arch Ration Mech Anal 209:869–997, 2013) proposed an extensive mathematical strategy to handle the problems studied numerically by Sone, Aoki and their group. The present paper offers an alternative, possibly simpler proof of one of the results discussed in Liu and Yu (2013).

Large deviations and central limit theorems for sequential and random systems of intermittent maps

We obtain large and moderate deviation estimates for both sequential and random compositions of intermittent maps. We also address the question of whether or not centering is necessary for the quenched central limit theorems obtained by Nicol, Török and Vaienti [Central limit theorems for sequential and random intermittent dynamical systems. Ergod. Th. & Dynam. Sys.38(3) (2018), 1127–1153] for random dynamical systems comprising intermittent maps. Using recent work of Abdelkader and Aimino [On the quenched central limit theorem for random dynamical systems. J. Phys. A 49(24) (2016), 244002] and Hella and Stenlund [Quenched normal approximation for random sequences of transformations. J. Stat. Phys.178(1) (2020), 1–37] we extend the results of Nicol, Török and Vaienti on quenched central limit theorems for centered observables over random compositions of intermittent maps: first by enlarging the parameter range over which the quenched central limit theorem holds; and second by showing that the variance in the quenched central limit theorem is almost surely constant (and the same as the variance of the annealed central limit theorem) and that centering is needed to obtain this quenched central limit theorem.

Invariance Principle for the Random Lorentz Gas—Beyond the Boltzmann-Grad Limit

We prove the invariance principle for a random Lorentz-gas particle in 3 dimensions under the Boltzmann-Grad limit and simultaneous diffusive scaling. That is, for the trajectory of a point-like particle moving among infinite-mass, hard-core, spherical scatterers of radius r, placed according to a Poisson point process of density $$\varrho $$ ϱ , in the limit $$\varrho \rightarrow \infty $$ ϱ → ∞ , $$r\rightarrow 0$$ r → 0 , $$\varrho r^{2}\rightarrow 1$$ ϱ r 2 → 1 up to time scales of order $$T=o(r^{-2}\left| {\log r}\right| ^{-2})$$ T = o ( r - 2 log r - 2 ) . To our knowledge this represents the first significant progress towards solving rigorously this problem in classical nonequilibrium statistical physics, since the groundbreaking work of Gallavotti (Phys Rev 185:308–322, 1969, Nota Interna Univ di Roma 358, 1970, Statistical mechanics. A short treatise. Theoretical and mathematical physics series, Springer, Berlin, 1999), Spohn (Commun Math Phys 60:277–290, 1978, Rev Mod Phys 52:569–611, 1980) and Boldrighini–Bunimovich–Sinai (J Stat Phys 32:477–501, 1983). The novelty is that the diffusive scaling of particle trajectory and the kinetic (Boltzmann-Grad) limit are taken simultaneously. The main ingredients are a coupling of the mechanical trajectory with the Markovian random flight process, and probabilistic and geometric controls on the efficiency of this coupling. Similar results have been earlier obtained for the weak coupling limit of classical and quantum random Lorentz gas, by Komorowski–Ryzhik (Commun Math Phys 263:277–323, 2006), respectively, Erdős–Salmhofer–Yau (Acta Math 200:211–277, 2008, Commun Math Phys 271:1–53, 2007). However, the following are substantial differences between our work and these ones: (1) The physical setting is different: low density rather than weak coupling. (2) The method of approach is different: probabilistic coupling rather than analytic/perturbative. (3) Due to (2), the time scale of validity of our diffusive approximation—expressed in terms of the kinetic time scale—is much longer and fully explicit.

Stationary currents in long-range interacting magnetic systems

We construct a solution for the 1d integro-differential stationary equation derived from a finite-volume version of the mesoscopic model proposed in Giacomin and Lebowitz (J. Stat. Phys. 87(1), 37–61, 1997). This is the continuous limit of an Ising spin chain interacting at long range through Kac potentials, staying in contact at the two edges with reservoirs of fixed magnetizations. The stationary equation of the model is introduced here starting from the Lebowitz-Penrose free energy functional defined on the interval [−ε− 1, ε− 1], ε > 0. Below the critical temperature, and for ε small enough, we obtain a solution that is no longer monotone when opposite in sign, metastable boundary conditions are imposed. Moreover, the mesoscopic current flows along the magnetization gradient. This can be considered as an analytic proof of the existence of diffusion along the concentration gradient in one-component systems undergoing a phase transition, a phenomenon generally known as uphill diffusion. In our proof uniqueness is lacking, and we have clues that the stationary solution obtained is not unique, as suggested by numerical simulations.

Effect of Self-Oscillation on Escape Dynamics of Classical and Quantum Open Systems

We study the effect of self-oscillation on the escape dynamics of classical and quantum open systems by employing the system-plus-environment-plus-interaction model. For a damped free particle (system) with memory kernel function expressed by Zwanzig (J. Stat. Phys. 9, 215 (1973)), which is originated from a harmonic oscillator bath (environment) of Debye type with cut-off frequency wd, ergodicity breakdown is found because the velocity autocorrelation function oscillates in cosine function for asymptotic time. The steady escape rate of such a self-oscillated system from a metastable potential exhibits nonmonotonic dependence on wd, which denotes that there is an optimal cut-off frequency makes it maximal. Comparing results in classical and quantum regimes, the steady escape rate of a quantum open system reduces to a classical one with wd decreasing gradually, and quantum fluctuation indeed enhances the steady escape rate. The effect of a finite number of uncoupled harmonic oscillators N on the escape dynamics of a classical open system is also discussed.