We prove an operator level limit for the circular Jacobi $beta$-ensemble. As a result, we characterize the counting function of the limit point process via coupled systems of stochastic differential equations. We also show that the normalized charact
eristic polynomials converge to a random analytic function, which we characterize via the joint distribution of its Taylor coefficients at zero and as the solution of a stochastic differential equation system. We also provide analogous results for the real orthogonal $beta$-ensemble.
We introduce a framework to study the random entire function $zeta_beta$ whose zeros are given by the Sine$_beta$ process, the bulk limit of beta ensembles. We present several equivalent characterizations, including an explicit power series represent
ation built from Brownian motion. We study related distributions using stochastic differential equations. Our function is a uniform limit of characteristic polynomials in the circular beta ensemble; we give upper bounds on the rate of convergence. Most of our results are new even for classical values of $beta$. We provide explicit moment formulas for $zeta$ and its variants, and we show that the Borodin-Strahov moment formulas hold for all $beta$ both in the limit and for circular beta ensembles. We show a uniqueness theorem for $zeta$ in the Cartwright class, and deduce some product identities between conjugate values of $beta$. The proofs rely on the structure of the Sine$_beta$ operator to express $zeta$ in terms of a regularized determinant.
The soft and hard edge scaling limits of $beta$-ensembles can be characterized as the spectra of certain random Sturm-Liouville operators. It has been shown that by tuning the parameter of the hard edge process one can obtain the soft edge process as
a scaling limit. We prove that this limit can be realized on the level of the corresponding random operators. More precisely, the random operators can be coupled in a way so that the scal
In the randomly-oriented Manhattan lattice, every line in $mathbb{Z}^d$ is assigned a uniform random direction. We consider the directed graph whose vertex set is $mathbb{Z}^d$ and whose edges connect nearest neighbours, but only in the direction fix
ed by the line orientations. Random walk on this directed graph chooses uniformly from the $d$ legal neighbours at each step. We prove that this walk is superdiffusive in two and three dimensions. The model is diffusive in four and more dimensions.
We provide a precise coupling of the finite circular beta ensembles and their limit process via their operator representations. We prove explicit bounds on the distance of the operators and the corresponding point processes. We also prove an estimate
on the beta-dependence in the $text{Sine}_{beta}$ process.
We show that Sine$_beta$, the bulk limit of the Gaussian $beta$-ensembles is the spectrum of a self-adjoint random differential operator [ fto 2 {R_t^{-1}} left[ begin{array}{cc} 0 &-tfrac{d}{dt} tfrac{d}{dt} &0 end{array} right] f, qquad f:[0,1)to m
athbb R^2, ] where $R_t$ is the positive definite matrix representation of hyperbolic Brownian motion with variance $4/beta$ in logarithmic time. The result connects the Montgomery-Dyson conjecture about the Sine$_2$ process and the non-trivial zeros of the Riemann zeta function, the Hilbert-Polya conjecture and de Branges attempt to prove the Riemann hypothesis. We identify the Brownian carousel as the Sturm-Liouville phase function of this operator. We provide similar operator representations for several other finite dimensional random ensembles and their limits: finite unitary or orthogonal ensembles, Hua-Pickrell ensembles and their limits, hard-edge $beta$-ensembles, as well as the Schrodinger point process. In this more general setting, hyperbolic Brownian motion is replaced by a random walk or Brownian motion on the affine group. Our approach provides a unified framework to study $beta$-ensembles that has so far been missing in the literature. In particular, we connect It^os classification of affine Brownian motions with the classification of limits of random matrix ensembles.
We give overcrowding estimates for the Sine_beta process, the bulk point process limit of the Gaussian beta-ensemble. We show that the probability of having at least n points in a fixed interval is given by $e^{-frac{beta}{2} n^2 log(n)+O(n^2)}$ as $
nto infty$. We also identify the next order term in the exponent if the size of the interval goes to zero.
We prove a version of the classical Dufresne identity for matrix processes. In particular, we show that the inverse Wishart laws on the space of positive definite r x r matrices can be realized by the infinite time horizon integral of M_t times its t
ranspose in which t -> M_t is a drifted Brownian motion on the general linear group. This solves a problem in the study of spiked random matrix ensembles which served as the original motivation for this result. Various known extensions of the Dufresne identity (and their applications) are also shown to have analogs in this setting. For example, we identify matrix valued diffusions built from M_t which generalize in a natural way the scalar processes figuring into the geometric Levy and Pitman theorems of Matsumoto and Yor.
We compute the fluctuation exponents for a solvable model of one-dimensional directed polymers in random environment in the intermediate regime. This regime corresponds to taking the inverse temperature to zero with the size of the system. The expone
nts satisfy the KPZ scaling relation and coincide with physical predictions. In the critical case, we recover the fluctuation exponents of the Cole-Hopf solution of the KPZ equation in equilibrium and close to equilibrium.
We study two one-parameter families of point processes connected to random matrices: the Sine_beta and Sch_tau processes. The first one is the bulk point process limit for the Gaussian beta-ensemble. For beta=1, 2 and 4 it gives the limit of the GOE,
GUE and GSE models of random matrix theory. In particular, for beta=2 it is a determinantal point process conjectured to have similar behavior to the critical zeros of the Riemann zeta-function. The second process can be obtained as the bulk scaling limit of the spectrum of certain discrete one-dimensional random Schrodinger operators. Both processes have asymptotically constant average density, in our normalization one expects close to lambda/(2pi) points in a large interval of length lambda. Our main results are large deviation principles for the average densities of the processes, essentially we compute the asymptotic probability of seeing an unusual average density in a large interval. Our approach is based on the representation of the counting functions of these processes using stochastic differential equations. We also prove path level large deviation principles for the arising diffusions. Our techniques work for the full range of parameter values. The results are novel even in the classical beta=1, 2 and 4 cases for the Sine_beta process. They are consistent with the existing rigorous results on large gap probabilities and confirm the physical predictions made using log-gas arguments.
