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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 consider a discrete-time stochastic growth model on the $d$-dimensional lattice with non-negative real numbers as possible values per site. The growth model describes various interesting examples such as oriented site/bond percolation, directed po
We review a recent approach to the mean-field limits in neural networks that takes into account the stochastic nature of input current and the uncertainty in synaptic coupling. This approach was proved to be a rigorous limit of the network equations
Products of $M$ i.i.d. non-Hermitian random matrices of size $N times N$ relate Gaussian fluctuation of Lyapunov and stability exponents in dynamical systems (finite $N$ and large $M$) to local eigenvalue universality in random matrix theory (finite
We study the averaged products of characteristic polynomials for the Gaussian and Laguerre $beta$-ensembles with external source, and prove Pearcey-type phase transitions for particular full rank perturbations of source. The phases are characterised
Under the Kolmogorov--Smirnov metric, an upper bound on the rate of convergence to the Gaussian distribution is obtained for linear statistics of the matrix ensembles in the case of the Gaussian, Laguerre, and Jacobi weights. The main lemma gives an