Introduction to KPZ

Introduction to Kpz

1. A physical introduction 2 1.1. KPZ/Stochastic Burgers/Scaling exponent 2 1.2. Physical derivation 3 1.3. Scaling 3 1.4. Formal invariance of Brownian motion 4 1.5. Dynamic scaling exponent 6 1.6. Renormalization of the nonlinear term 6 1.7. Cutoff KPZ models 7 1.8. Hopf-Cole solutions 8 1.9. Directed polymers in a random environment 11 1.10. Fluctuation breakthroughs of 1999 12 1.11. The Air...

An Introduction to Sociolinguistics

Introduction to circular data

In many diverse scientific fields, the measurements are directions. For instance, a biologist may be measuring the direction of flight of a bird or the orientation of an animal. A series of such observations is called ”directional data”. Since a direction has no magnitude, these can be conveniently represented as points on the circumference of a unit circle centered at the origin or as unit ...

Introduction to Ethics

  The issues involved in this article are mainly concentrated on the origins of ethics or Moral ethics as an indepent science It has been said that this area of science was founded by Greek Philosophers after a great crisis and debates between such thinkers as Socrates, Plato and Aristotle on one side and  study of other cuitures and moral traditions, especially the Chinese and Hindu tradition...

Liouville quantum gravity and KPZ

Consider a bounded planar domain D, an instance h of the Gaussian free field on D, with Dirichlet energy (2π)−1 ∫ D ∇h(z) · ∇h(z)dz, and a constant 0 ≤ γ < 2. The Liouville quantum gravity measure on D is the weak limit as ε → 0 of the measures ε 2/2eγhε(z)dz, where dz is Lebesgue measure on D and hε(z) denotes the mean value of h on the circle of radius ε centered at z. Given a random (or dete...