Uniqueness of fixed point of a two-dimensional map obtained as a generalization of the renormalization group map associated to the self-avoiding paths on gaskets

Let $W(x,y) = a x^3 + b x^4 + f_5 x^5 + f_6 x^6 + (3 a x^2)^2 y + g_5 x^5 y + h_3 x^3 y^2 + h_4 x^4 y^2 + n_3 x^3 y^3 + a_{24} x^2 y^4 + a_{05} y^5 + a_{15} x y^5 + a_{06} y^6$, and $X=\frac{\partial W}{\partial x}$, $Y=\frac{\partial W}{\partial y}$, where the coefficients are non-negative constants, with $a>0$, such that $X^{2}(x,x^{2})-Y(x,x^{2})$ is a polynomial of $x$ with non-negative coefficients. Examples of the 2 dimensional map $\Phi: (x,y)\mapsto (X(x,y),Y(x,y))$ satisfying the conditions are the renormalization group (RG) map (modulo change of variables) for the restricted self-avoiding paths on the 3 and 4 dimensional pre-gaskets. We prove that there exists a unique fixed point $(x_f,y_f)$ of $\Phi$ in the invariant set $\{(x,y)\in R^2\mid x^2\ge y\}\setminus\{0\}$.Comment: LaTeX2e, 12 pages, no figure

Stochastic ranking process with space-time dependent intensities

We consider the stochastic ranking process with space-time dependent jump rates for the particles. The process is a simplified model of the time evolution of the rankings such as sales ranks at online bookstores. We prove that the joint empirical distribution of jump rate and scaled position converges almost surely to a deterministic distribution, and also the tagged particle processes converge almost surely, in the infinite particle limit. The limit distribution is characterized by a system of inviscid Burgers-like integral-partial differential equations with evaporation terms, and the limit process of a tagged particle is a motion along a characteristic curve of the differential equations except at its Poisson times of jumps to the origin

Mathematical Derivation of Chiral Anomaly in Lattice Gauge Theory with Wilson's Action

Chiral U(1) anomaly is derived with mathematical rigor for a Euclidean fermion coupled to a smooth external U(1) gauge field on an even dimensional torus as a continuum limit of lattice regularized fermion field theory with the Wilson term in the action. The present work rigorously proves for the first time that the Wilson term correctly reproduces the chiral anomaly.Comment: 33 pages, LaTe

Existence of an infinite particle limit of stochastic ranking process

We study a stochastic particle system which models the time evolution of the ranking of books by online bookstores (e.g., Amazon). In this system, particles are lined in a queue. Each particle jumps at random jump times to the top of the queue, and otherwise stays in the queue, being pushed toward the tail every time another particle jumps to the top. In an infinite particle limit, the random motion of each particle between its jumps converges to a deterministic trajectory. (This trajectory is actually observed in the ranking data on web sites.) We prove that the (random) empirical distribution of this particle system converges to a deterministic space-time dependent distribution. A core of the proof is the law of large numbers for {\it dependent} random variables

Stochastic ranking process with time dependent intensities

We consider the stochastic ranking process with the jump times of the particles determined by Poisson random measures. We prove that the joint empirical distribution of scaled position and intensity measure converges almost surely in the infinite particle limit. We give an explicit formula for the limit distribution and show that the limit distribution function is a unique global classical solution to an initial value problem for a system of a first order non-linear partial differential equations with time dependent coefficients