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Register a new userScaling limit for trap models on $mathbb{Z}^d$
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We give the ``quenched scaling limit of Bouchauds trap model in ${dge 2}$. This scaling limit is the fractional-kinetics process, that is the time change of a $d$-dimensional Brownian motion by the inverse of an independent $alpha$-stable subordinator.
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Some stochastic systems are particularly interesting as they exhibit critical behavior without fine-tuning of a parameter, a phenomenon called self-organized criticality. In the context of driven-dissipative steady states, one of the main models is that of Activated Random Walks. Long-range effects intrinsic to the conservative dynamics and lack of a simple algebraic structure cause standard tools and techniques to break down. This makes the mathematical study of this model remarkably challenging. Yet, some exciting progress has been made in the last ten years, with the development of a framework of tools and methods which is finally becoming more structured. In these lecture notes we present the existing results and reproduce the techniques developed so far.
We investigate the percolation phase transition for level sets of the Gaussian free field on $mathbb{Z}^d$, with $dgeqslant 3$, and prove that the corresponding critical parameter $h_*(d)$ is strictly positive for all $dgeqslant3$, thus settling an open question from arXiv:1202.5172. In particular, this implies that the sign clusters of the Gaussian free field percolate on $mathbb{Z}^d$, for all $dgeqslant 3$. Among other things, our construction of an infinite cluster above small, but positive level $h$ involves random interlacements at level $u>0$, a random subset of $mathbb{Z}^d$ with desirable percolative properties, introduced in arXiv:0704.2560 in a rather different context, a certain Dynkin-type isomorphism theorem relating random interlacements to the Gaussian free field, see arXiv:1111.4818, and a recent coupling from arXiv:1402.0298 of these two objects, lifted to a continuous metric graph structure over $mathbb{Z}^d$.
We prove almost sure Euler hydrodynamics for a large class of attractive particle systems on $Z$ starting from an arbitrary initial profile. We generalize earlier works by Seppalainen (1999) and Andjel et al. (2004). Our constructive approach requires new ideas since the subadditive ergodic theorem (central to previous works) is no longer effective in our setting.
We consider a class of continuous-time stochastic growth models on $d$-dimensional lattice with non-negative real numbers as possible values per site. We remark that the diffusive scaling limit proven in our previous work [Nagahata, Y., Yoshida, N.: Central Limit Theorem for a Class of Linear Systems, Electron. J. Probab. Vol. 14, No. 34, 960--977. (2009)] can be extended to wider class of models so that it covers the cases of potlatch/smoothing processes.
A two-type version of the frog model on $mathbb{Z}^d$ is formulated, where active type $i$ particles move according to lazy random walks with probability $p_i$ of jumping in each time step ($i=1,2$). Each site is independently assigned a random number of particles. At time 0, the particles at the origin are activated and assigned type 1 and the particles at one other site are activated and assigned type 2, while all other particles are sleeping. When an active type $i$ particle moves to a new site, any sleeping particles there are activated and assigned type $i$, with an arbitrary tie-breaker deciding the type if the site is hit by particles of both types in the same time step. We show that the event $G_i$ that type $i$ activates infinitely many particles has positive probability for all $p_1,p_2in(0,1]$ ($i=1,2$). Furthermore, if $p_1=p_2$, then the types can coexist in the sense that $mathbb{P}(G_1cap G_2)>0$. We also formulate several open problems. For instance, we conjecture that, when the initial number of particles per site has a heavy tail, the types can coexist also when $p_1 eq p_2$.
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