arXiv subjects

F. Martinelli

Publications and source records attributed to F. Martinelli.

27 records · Page 2Linked to original sources

arXiv2013

Kinetically constrained spin models on trees

We analyze kinetically constrained 0-1 spin models (KCSM) on rooted and unrooted trees of finite connectivity. We focus in particular on the class of Friedrickson-Andersen models FA-jf and on an oriented version of them. These tree models are particularly relevant in physics literature since some of them undergo an ergodicity breaking transition with the mixed first-second order character of the glass transition. Here we first identify the ergodicity regime and prove that the critical density for FA-jf and OFA-jf models coincide with that of a suitable bootstrap percolation model. Next we prove for the first time positivity of the spectral gap in the whole ergodic regime via a novel argument based on martingales ideas. Finally, we discuss how this new technique can be generalized to analyze KCSM on the regular lattice $\mathbb{Z}^d$.

math.PR↗

arXiv2012

Study of the Cataclysmic Variable 1RXS J174320.1-042953

We report on the photometric analysis of the poorly studied cataclysmic variable in Ophiuchus 1RXS J174320.1-042953 (DDE 11). Results of the monitoring with the Bradford Robotic Telescope (BRT) are presented, as well as the time-resolved photometry obtained at Montecatini Astronomical Centre. The long-term behavior of J1743-0429 shows it is a magnetic cataclysmic variable (polar) with the large difference between the low and high states. The orbital period obtained from our observations is 0.0866d, or 2.08hr, close to the lower boundary of the period gap of cataclysmic variables.

astro-ph.SR↗

arXiv2011

Linear feedback control of transient energy growth and control performance limitations in subcritical plane Poiseuille flow

Suppression of the transient energy growth in subcritical plane Poiseuille flow via feedback control is addressed. It is assumed that the time derivative of any of the velocity components can be imposed at the walls as control input, and that full-state information is available. We show that it is impossible to design a linear state-feedback controller that leads to a closed-loop flow system without transient energy growth. In a subsequent step, full-state feedback controllers -- directly targeting the transient growth mechanism -- are designed, using a procedure based on a Linear Matrix Inequalities approach. The performance of such controllers is analyzed first in the linear case, where comparison to previously proposed linear-quadratic optimal controllers is made; further, transition thresholds are evaluated via Direct Numerical Simulations of the controlled three-dimensional Poiseuille flow against different initial conditions of physical interest, employing different velocity components as wall actuation. The present controllers are effective in increasing the transition thresholds in closed loop, with varying degree of performance depending on the initial condition and the actuation component employed.

physics.flu-dyn↗

arXiv2010

Aging through hierarchical coalescence in the East model

We rigorously analyze the low temperature non-equilibrium dynamics of the East model, a special example of a one dimensional oriented kinetically constrained particle model, when the initial distribution is different from the reversible one and for times much smaller than the global relaxation time. This setting has been intensively studied in the physics literature to analyze the slow dynamics which follows a sudden quench from the liquid to the glass phase. In the limit of zero temperature (i.e. a vanishing density of vacancies) and for initial distributions such that the vacancies form a renewal process we prove that the density of vacancies, the persistence function and the two-time autocorrelation function behave as staircase functions with several plateaux. Furthermore the two-time autocorrelation function displays an aging behavior. We also provide a sharp description of the statistics of the domain length as a function of time, a domain being the interval between two consecutive vacancies. When the initial renewal process has finite mean our results confirm (and generalize) previous findings of the physicists for the restricted case of a product Bernoulli measure. However we show that a different behavior appears when the initial domain distribution is in the attraction domain of a $α$-stable law. All the above results actually follow from a more general result which says that the low temperature dynamics of the East model is very well described by that of a certain hierarchical coalescence process, a probabilistic object which can be viewed as a hierarchical sequence of suitably linked coalescence processes and whose asymptotic behavior has been recently studied in [14].

math-ph↗

arXiv2009

On the mixing time of the 2D stochastic Ising model with "plus" boundary conditions at low temperature

We consider the Glauber dynamics for the 2D Ising model in a box of side L, at inverse temperature $β$ and random boundary conditions $τ$ whose distribution P either stochastically dominates the extremal plus phase (hence the quotation marks in the title) or is stochastically dominated by the extremal minus phase. A particular case is when P is concentrated on the homogeneous configuration identically equal to + (equal to -). For $β$ large enough we show that for any $ε$ there exists $c=c(β,ε)$ such that the corresponding mixing time $T_{mix}$ satisfies $\lim_{L\to\infty}P(T_{mix}> \exp({cL^ε})) =0$. In the non-random case $τ\equiv +$ (or $τ\equiv -$), this implies that $T_{mix}< \exp({cL^ε})$. The same bound holds when the boundary conditions are all + on three sides and all - on the remaining one. The result, although still very far from the expected Lifshitz behaviour $T_{mix}=O(L^2)$, considerably improves upon the previous known estimates of the form $T_{mix}\le \exp({c L^{1/2 + ε}})$. The techniques are based on induction over length scales, combined with a judicious use of the so-called "censoring inequality" of Y. Peres and P. Winkler, which in a sense allows us to guide the dynamics to its equilibrium measure.

math.PR↗

arXiv2006

Relaxation time of $L$-reversal chains and other chromosome shuffles

We prove tight bounds on the relaxation time of the so-called $L$-reversal chain, which was introduced by R. Durrett as a stochastic model for the evolution of chromosome chains. The process is described as follows. We have $n$ distinct letters on the vertices of the ${n}$-cycle (${\mathbb{Z}}$ mod $n$); at each step, a connected subset of the graph is chosen uniformly at random among all those of length at most $L$, and the current permutation is shuffled by reversing the order of the letters over that subset. We show that the relaxation time $τ(n,L)$, defined as the inverse of the spectral gap of the associated Markov generator, satisfies $τ(n,L)=O(n\vee \frac{n^3}{L^3})$. Our results can be interpreted as strong evidence for a conjecture of R. Durrett predicting a similar behavior for the mixing time of the chain.

math.PR↗

arXiv2003

Hydrodynamic limit of a disordered lattice gas

We consider a model of lattice gas dynamics in the d-dimensional cubic lattice in the presence of disorder. If the particle interaction is only mutual exclusion and if the disorder field is given by i.i.d. bounded random variables, we prove the almost sure existence of the hydrodynamical limit in dimension d>2. The limit equation is a non linear diffusion equation with diffusion matrix characterized by a variational principle.

math.PR↗

arXiv2002

Some New Results on the Kinetic Ising Model in a Pure Phase

We consider a general class of Glauber dynamics reversible with respect to the standard Ising model in $\bbZ^d$ with zero external field and inverse temperature $\gb$ strictly larger than the critical value $\gb_c$ in dimension 2 or the so called ``slab threshold'' $\hat \b_c$ in dimension $d \geq 3$. We first prove that the inverse spectral gap in a large cube of side $N$ with plus boundary conditions is, apart from logarithmic corrections, larger than $N$ in $d=2$ while the logarithmic Sobolev constant is instead larger than $N^2$ in any dimension. Such a result substantially improves over all the previous existing bounds and agrees with a similar computations obtained in the framework of a one dimensional toy model based on mean curvature motion. The proof, based on a suggestion made by H.T. Yau some years ago, explicitly constructs a subtle test function which forces a large droplet of the minus phase inside the plus phase. The relevant bounds for general $d\ge 2$ are then obtained via a careful use of the recent $\bbL^1$--approach to the Wulff construction. Finally we prove that in $d=2$ the probability that two independent initial configurations, distributed according to the infinite volume plus phase and evolving under any coupling, agree at the origin at time $t$ is bounded from below by a stretched exponential $\exp(-\sqrt{t})$, again apart from logarithmic corrections. Such a result should be considered as a first step toward a rigorous proof that, as conjectured by Fisher and Huse some years ago, the equilibrium time auto-correlation of the spin at the origin decays as a stretched exponential in $d=2$.

math-ph↗

arXiv2001

Droplet Motion for the Conservative 2D Ising Lattice Gas Dynamics below the Critical Temperature

We consider the 2D Ising lattice gas in a square of side L with free boundary conditions, temperature below the critical one and particle density slightly above the density of the vapor phase. Typical configurations consist of a quarter of a Wulff droplet of the liquid phase centered at one of the corners of the given square. We then introduced a reversible Markovian spin exchange dynamics, also known as Kawasaki dynamics, on the configuration space and we discuss the heuristics of the transition of a bubble of the liquid phase from one corner to another. We then present some numerical evidence suggesting that the preferred mechanism to make the transition is via evaporation of the original bubble and simultaneous reconstruction of a new bubble around a new corner.

cond-mat.stat-mech↗