arXiv subjects

M. Katori

Publications and source records attributed to M. Katori.

3 recordsLinked to original sources

arXiv2001

Scaling Limit of Vicious Walkers, Gaussian Random Matrix Ensembles, and Dyson Brownian Motions

We study systems of interacting Brownian particles in one dimension constructed as the diffusion scaling limits of Fisher's vicious walk models. We define two types of nonintersecting Brownian motions, in which we impose no condition (resp. nonintersecting condition forever) in the future for the first-type (resp. second-type). It is shown that, when all particles start from the origin, their positions at time 1 in the first-type (resp. at time 1/2 in the second-type) process are identically distributed with the eigenvalues of Gaussian orthogonal (resp.unitary) random matrices. The second-type process is described by the stochastic differential equations of the Dyson-type Brownian motions with repulsive two-body forces proportional to the inverse of distances. The present study demonstrates that the spatio-temporal coarse graining of random walk models with contact interactions can provide many-body systems with long-range interactions.

cond-mat.stat-mech↗

arXiv2001

Scaling Limit of Vicious Walkers, Schur Function, and Gaussian Random Matrix Ensemble

We consider the diffusion scaling limit of the vicious walkers and derive the time-dependent spatial-distribution function of walkers. The dependence on initial configurations of walkers is generally described by using the symmetric polynomials called the Schur functions. In the special case in the scaling limit that all walkers are started from the origin, the probability density is simplified and it shows that the positions of walkers on the real axis at time one is identically distributed with the eigenvalues of random matrices in the Gaussian orthogonal ensemble. Since the diffusion scaling limit makes the vicious walkers converge to the nonintersecting Brownian motions in distribution, the present study will provide a new method to analyze intersection problems of Brownian motions in one-dimension.

cond-mat.stat-mech↗

arXiv1995

Some aspects of the critical behavior of the Two-Neighbor Stochastic Cellular Automata

Using Pade approximations and Monte Carlo simulations, we study the phase diagram of the Two-Neighbor Stochastic Cellular Automata, which have two parameters $p_{1}$ and $p_{2}$ and include the mixed site-bond directed percolation (DP) as a special case. The phase transition line $p_{1}=p_{1 {\rm c}}(p_{2})$ has endpoints at $(p_{1}, p_{2})=(1/2,1)$ and at (0.8092, 0). The former point (1/2,1) is a special point at which Compact DP transition occurs and its critical exponents are known exactly. Results of time-dependent simulation show that in the whole range of parameters, excluding this point (1/2,1), the system belongs to the DP universality class. It is first shown that the shape of the phase transition line near this special point has, asymptotically, a parabolic shape, {\it i.e.}, $p_{1{\rm c}}(p_{2})-1/2 \sim (1-p_{2})^θ$ with $θ= 1/2$ for $0 < 1-p_{2} \ll 1$. We use the Monte Carlo data to assess the accuracy of rigorous bounds for the line recently reported by Liggett and by Katori and Tsukahara. It is also shown that outside the vicinity of the special point (1/2,1), the curve is well approximated by an interpolation formula, similar to the one proposed by Yanuka and Englman.

cond-mat↗