General \(\beta\)-ensembles or log-gas in dimension 1
A general \(\beta\)-ensemble is a system of \(N\) particles with positions \(\lambda_1,\dots,\lambda_N\) on the real line,
chosen according to the following measure
\[
\frac{1}{Z_{N,\beta}}
\prod_{1 \leq i < j \leq N} \lvert \lambda_i - \lambda_j \rvert^\beta
e^{- \frac{\beta N}{2} \sum_{k=1}^N V(\lambda_k)}
\mathrm{d} \lambda_1 \cdots \mathrm{d} \lambda_N,
\]
where \(V : \mathbb{R} \to \mathbb{R}\) is a confining potential, \(\beta > 0\) is the inverse temperature
and \(Z_{N,\beta}\) is a normalizing constant.
This means that each particle is affected by the potential \(V\) and by repulsive logarthmic interactions with the other particles.
The particular case \(V(x) = x^2/2\) corresponds to the Gaussian \(\beta\)-ensembles.
In that case, the particules \(\lambda_1,\dots,\lambda_N\) can be seen as the eigenvalues of a tridiagonal matrix with random entries,
whose distributions depend on \(\beta\).
The asymptotic distribution of particles is the semi-circle distribution on \([-2,2]\).
The pictures below are obtained in that case.
This family of models include the extensively studied GOE (\(\beta=1\)), GUE (\(\beta=2\)) and GSE (\(\beta=4\)).
The log-characteristic polynomial
In a joint work with Paul Bourgade and Krishnan Mody, we study the logarithm of the characteristic polynomial of general \(\beta\)-ensembles.
More precisely, we consider the following centered version of the log-characteristic polynomial
\[
X_N(x) = \sum_{k=1}^N \log(x-\lambda_k) - N \int_{\mathbb{R}} \log(x-\lambda) \mu_{\mathrm{eq}} (\mathrm{d} \lambda),
\]
where we consider a branch of the logarithm such that \(\log x = \log \lvert x \rvert + i \pi \mathbb{1}_{x<0} \) for \(x \in \mathbb{R}^*\).
We prove that, in the bulk of the spectrum, the real and imaginary parts of \(X_N\) behave asymptotically as Gaussian log-correlated fields,
with pointwise variance \(\log N\).
Moreover, the real and imaginary parts are asymptotically independent at scale \(\sqrt{\log N}\).
The pictures below represent the real and imaginary parts of \(X_N\) in the case of the Gaussian \(\beta\)-ensemble with \(\beta=3\).
The real part is pictured on the left: note that there is a divergence at each particle, even if this does not appear in the typical behavior.
The imaginary part is on the right: since the imaginary part of the logarithm is a step function on the real line, this represents
the recentered counting function of particles.
\(N=100\)
\(N=500\)
\(N=2000\)
Note that the real part does not look symmetric, due to the logarithmic divergences at each particle.
However, this behavior is not typical and, for this reason, the limiting distribution can still be gaussian.
In the pictures below, the real part of \(X_N\) is represented again, but with a fixed discretization of the interval by \(5N\) points.
By doing so, one cannot see the divergences anymore.
