Harmonically confined Riesz gas in one dimension
A talk in the Seminar Zufallsmatrizen series by
Satya Majumdar from LPTMS, Universite de Paris-Sud
| Abstract: | I will discuss one dimensional Riesz gas of N particles confined in a
harmonic potential. The interaction between any pair of particles at
positions $xxii$ and $xxjj$ is repulsive and behaves as \sgn(k) $| xxii − xxjj | -kk$
for $ii \neq jj$, where $kk>-2$. For $k = -1$, this model represents the one
dimensional one component plasma, for $kk \rightarrow 0$ it represents Dyson's log-gas
that appears in random matrix theory and for $kk = 2$, it represents the
classical Calogero-Moser model. We will first compute the average density in
the large N limit explicitly for all $kk > -2$. Next, we will compute the
exact average density (large N limit) in the presence of a hard wall at
$xx = ww$. Finally, I will discuss the statistics of the position of the
rightmost particle in the gas, and will compute the explicit large
deviation functions of its distribution. We will see that the left tail
exhibits a third order phase transition for all $kk > -2$. Please contact Gernot Akemann (akemann@physik.uni-bielefeld.de) for details regarding access. Within the CRC this talk is associated to the project(s): C6 |