310 research outputs found

### 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

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