A weakness in strong localization for Sinai's walk

Sinai's walk is a recurrent one-dimensional nearest-neighbor random walk in random environment. It is known for a phenomenon of strong localization, namely, the walk spends almost all time at or near the bottom of deep valleys of the potential. Our main result shows a weakness of this localization phenomenon: with probability one, the zones where the walk stays for the most time can be far away from the sites where the walk spends the most time. In particular, this gives a negative answer to a problem of Erd\H{o}s and R\'{e}v\'{e}sz [Mathematical Structures--Computational Mathematics--Mathematical Modelling 2 (1984) 152--157], originally formulated for the usual homogeneous random walk.Comment: Published at http://dx.doi.org/10.1214/009117906000000863 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Poisson-Dirichlet statistics for the extremes of a log-correlated Gaussian field

We study the statistics of the extremes of a discrete Gaussian field with logarithmic correlations at the level of the Gibbs measure. The model is defined on the periodic interval $[0,1]$, and its correlation structure is nonhierarchical. It is based on a model introduced by Bacry and Muzy [Comm. Math. Phys. 236 (2003) 449-475] (see also Barral and Mandelbrot [Probab. Theory Related Fields 124 (2002) 409-430]), and is similar to the logarithmic Random Energy Model studied by Carpentier and Le Doussal [Phys. Rev. E (3) 63 (2001) 026110] and more recently by Fyodorov and Bouchaud [J. Phys. A 41 (2008) 372001]. At low temperature, it is shown that the normalized covariance of two points sampled from the Gibbs measure is either $0$ or $1$. This is used to prove that the joint distribution of the Gibbs weights converges in a suitable sense to that of a Poisson-Dirichlet variable. In particular, this proves a conjecture of Carpentier and Le Doussal that the statistics of the extremes of the log-correlated field behave as those of i.i.d. Gaussian variables and of branching Brownian motion at the level of the Gibbs measure. The method of proof is robust and is adaptable to other log-correlated Gaussian fields.Comment: Published in at http://dx.doi.org/10.1214/13-AAP952 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Aging and quenched localization for one-dimensional random walks in random environment in the sub-ballistic regime

We consider transient one-dimensional random walks in random environment with zero asymptotic speed. An aging phenomenon involving the generalized Arcsine law is proved using the localization of the walk at the foot of "valleys" of height $\log t$. In the quenched setting, we also sharply estimate the distribution of the walk at time $t$.Comment: 24 pages, accepted for publication in "Bulletin de la SMF

Limit laws for transient random walks in random environment on \z

We consider transient random walks in random environment on \z with zero asymptotic speed. A classical result of Kesten, Kozlov and Spitzer says that the hitting time of the level $n$ converges in law, after a proper normalization, towards a positive stable law, but they do not obtain a description of its parameter. A different proof of this result is presented, that leads to a complete characterization of this stable law. The case of Dirichlet environment turns out to be remarkably explicit.Comment: 31 pages, accepted for publication in "Annales de l'Institut Fourier

A probabilistic representation of constants in Kesten's renewal theorem

The aims of this paper are twofold. Firstly, we derive some probabilistic representation for the constant which appears in the one-dimensional case of Kesten's renewal theorem. Secondly, we estimate the tail of some related random variable which plays an essential role in the description of the stable limit law of one-dimensional transient sub-ballistic random walks in random environment.Comment: 27 page

Universality and Sharpness in Absorbing-State Phase Transitions

We consider the Activated Random Walk model in any dimension with any sleep rate and jump distribution and ergodic initial state. We show that the stabilization properties depend only on the average density of particles, regardless of how they are initially located on the lattice

Pengaruh Lama Penggorengan Terhadap Kadar Vitamin C Dan Daya Terima Keripik Pepaya Yang Digoreng Menggunakan Metode Kovensional Dan Vakum

Pepaya merupakan buah musiman dengan masa simpan relatif pendek sehingga perlu usaha untuk memperpanjang masa simpan. Salah satu upaya untuk memperpanjang masa simpan pepaya yaitu dengan mengolahnya menjadi keripik . Penelitian ini bertujuan untuk mengetahui lama penggorengan terhadap kadar vitamin C dan daya terima keripik pepaya yang digoreng menggunakan penggorengan kovensional dan vakum Jenis penelitian ini menggunakan desain ekperimental, metode penggorengan kovensional dengan suhu 1200C (15 detik, 20 detik, 25 detik), vakum dengan suhu 750C (30 menit, 40 menit dan 50 menit) dan pengujian penerimaan sensorik. Percobaan menggunakan rancangan acak lengkap (RAL) dengan 2 kali perlakuan dan 3 kali ulangan. Analisis data menggunakan uji ANOVA satu arah dengan tingkat kepercayaan 95% menggunakan program SPSS versi 16 apabila p=value > 0,05 maka tidak terdapat pengaruh, apabila p=value <0,05 maka terdapat pengaruh atau perbedaan. Rata-rata kadar vitamin C penggorengan kovensional nilai p 0,000 (0,000<0,05) menunjukkan ada pengaruh lama penggorengan dan rata-rata kadar vitamin C pada penggorengan vakum nilai p 0,002 (0,002<0,05) menunjukan ada pengaruh lama penggorengan. Daya terima keripik pepaya menggunakan penggorengan kovensional nilai signifikan p=0,000 (0,000<0,05) menunjukkan ada perbedaan daya terima kerenyahan terhadap lama penggorengan keripik pepaya menggunakan penggorengan kovensional. Daya terima keripik pepaya menggunakan penggorengan vakum nilai p<0,05 menunjukkan ada perbedaan daya terima kerenyahan dan kesukaan dengan nilai p<0,05. Terdapat perbedaan pengaruh lama penggorengan terhadap kadar vitamin C dan daya terima keripik pepaya yang digoreng menggunakan penggorengan kovensional dan vakum. Saran dari penelitian ini adalah penggorengan keripik pepaya kovensional dengan suhu 120ºC selama 15 detik dan penggorengan vakum dengan suhu 75ºC selama 50 menit

Stable fluctuations for ballistic random walks in random environment on Z

We consider transient random walks in random environment on Z in the positive speed (ballistic) and critical zero speed regimes. A classical result of Kesten, Kozlov and Spitzer proves that the hitting time of level $n$, after proper centering and normalization, converges to a completely asymmetric stable distribution, but does not describe its scale parameter. Following a previous article by three of the authors, where the (non-critical) zero speed case was dealt with, we give a new proof of this result in the subdiffusive case that provides a complete description of the limit law. The case of Dirichlet environment turns out to be remarkably explicit

Scaling limit and aging for directed trap models

We consider one-dimensional directed trap models and suppose that the trapping times are heavy-tailed. We obtain the inverse of a stable subordinator as scaling limit and prove an aging phenomenon expressed in terms of the generalized arcsine law. These results confirm the status of universality described by Ben Arous and \v{C}ern\'y for a large class of graphs.Comment: 16 pages, accepted for publication in "Markov processes and Related Fields