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Register a new userUniversality and Sharpness in Absorbing-State Phase Transitions
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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.
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Consider $Ntimes N$ symmetric one-dimensional random band matrices with general distribution of the entries and band width $W geq N^{3/4+varepsilon}$ for any $varepsilon>0$. In the bulk of the spectrum and in the large $N$ limit, we obtain the following results. (i) The semicircle law holds up to the scale $N^{-1+varepsilon}$ for any $varepsilon>0$. (ii) The eigenvalues locally converge to the point process given by the Gaussian orthogonal ensemble at any fixed energy. (iii) All eigenvectors are delocalized, meaning their ${rm L}^infty$ norms are all simultaneously bounded by $N^{-frac{1}{2}+varepsilon}$ (after normalization in ${rm L}^2$) with overwhelming probability, for any $varepsilon>0$. (iv )Quantum unique ergodicity holds, in the sense that the local ${rm L}^2$ mass of eigenvectors becomes equidistributed with overwhelming probability. We extend the mean-field reduction method cite{BouErdYauYin2017}, which required $W=Omega(N)$, to the current setting $W ge N^{3/4+varepsilon}$. Two new ideas are: (1) A new estimate on the generalized resolvent of band matrices when $W geq N^{3/4+varepsilon}$. Its proof, along with an improved fluctuation average estimate, will be presented in parts 2 and 3 of this series cite {BouYanYauYin2018,YanYin2018}. (2) A strong (high probability) version of the quantum unique ergodicity property of random matrices. For its proof, we construct perfect matching observables of eigenvector overlaps and show they satisfying the eigenvector moment flow equation cite{BouYau2017} under the matrix Brownian motions.
The effects of quenched disorder on nonequilibrium phase transitions in the directed percolation universality class are revisited. Using a strong-disorder energy-space renormalization group, it is shown that for any amount of disorder the critical behavior is controlled by an infinite-randomness fixed point in the universality class of the random transverse-field Ising models. The experimental relevance of our results are discussed.
We consider a class of nonlinear mappings $mathsf{F}_{A,N}$ in $mathbb{R}^N$ indexed by symmetric random matrices $Ainmathbb{R}^{Ntimes N}$ with independent entries. Within spin glass theory, special cases of these mappings correspond to iterating the TAP equations and were studied by Bolthausen [Comm. Math. Phys. 325 (2014) 333-366]. Within information theory, they are known as approximate message passing algorithms. We study the high-dimensional (large $N$) behavior of the iterates of $mathsf{F}$ for polynomial functions $mathsf{F}$, and prove that it is universal; that is, it depends only on the first two moments of the entries of $A$, under a sub-Gaussian tail condition. As an application, we prove the universality of a certain phase transition arising in polytope geometry and compressed sensing. This solves, for a broad class of random projections, a conjecture by David Donoho and Jared Tanner.
We consider a simple discrete-time Markov chain with values in $[0,infty)^{Z^d}$. The Markov chain describes various interesting examples such as oriented percolation, directed polymers in random environment, time discretizations of binary contact path process and the voter model. We study the phase transition for the growth rate of the total number of particles in this framework. The main results are roughly as follows: If $d ge 3$ and the Markov chain is not too random, then, with positive probability, the growth rate of the total number of particles is of the same order as its expectation. If on the other hand, $d=1,2$, or the Markov chain is random enough, then the growth rate is slower than its expectation. We also discuss the above phase transition for the dual processes and its connection to the structure of invariant measures for the Markov chain with proper normalization.
We study absorbing phase transitions in systems of branching annihilating random walkers and pair contact process with diffusion on a one dimensional ring, where the walkers hop to their nearest neighbor with a bias $epsilon$. For $epsilon=0$, three universality classes: directed percolation (DP), parity conserving (PC) and pair contact process with diffusion (PCPD) are typically observed in such systems. We find that the introduction of $epsilon$ does not change the DP universality class but alters the other two universality classes. For non-zero $epsilon$, the PCPD class crosses over to DP and the PC class changes to a new universality class.
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