Metastability of reversible condensed zero range processes on a finite set

Let r: S\times S\to \bb R_+ be the jump rates of an irreducible random walk on a finite set $S$, reversible with respect to some probability measure $m$. For $\alpha >1$, let g: \bb N\to \bb R_+ be given by $g(0)=0$, $g(1)=1$, $g(k) = (k/k-1)^\alpha$, $k\ge 2$. Consider a zero range process on $S$ in which a particle jumps from a site $x$, occupied by $k$ particles, to a site $y$ at rate $g(k) r(x,y)$. Let $N$ stand for the total number of particles. In the stationary state, as $N\uparrow\infty$, all particles but a finite number accumulate on one single site. We show in this article that in the time scale $N^{1+\alpha}$ the site which concentrates almost all particles evolves as a random walk on $S$ whose transition rates are proportional to the capacities of the underlying random walk

Reduction formula for fermion loops and density correlations of the 1D Fermi gas

Fermion N-loops with an arbitrary number of density vertices N > d+1 in d spatial dimensions can be expressed as a linear combination of (d+1)-loops with coefficients that are rational functions of external momentum and energy variables. A theorem on symmetrized products then implies that divergencies of single loops for low energy and small momenta cancel each other when loops with permuted external variables are summed. We apply these results to the one-dimensional Fermi gas, where an explicit formula for arbitrary N-loops can be derived. The symmetrized N-loop, which describes the dynamical N-point density correlations of the 1D Fermi gas, does not diverge for low energies and small momenta. We derive the precise scaling behavior of the symmetrized N-loop in various important infrared limits.Comment: 14 pages, to be published in Journal of Statistical Physic

A Proof of the G\"ottsche-Yau-Zaslow Formula

Let S be a complex smooth projective surface and L be a line bundle on S. G\"ottsche conjectured that for every integer r, the number of r-nodal curves in |L| is a universal polynomial of four topological numbers when L is sufficiently ample. We prove G\"ottsche's conjecture using the algebraic cobordism group of line bundles on surfaces and degeneration of Hilbert schemes of points. In addition, we prove the the G\"ottsche-Yau-Zaslow Formula which expresses the generating function of the numbers of nodal curves in terms of quasi-modular forms and two unknown series.Comment: 29 page