Let $mathcal{T}$ be a rooted tree endowed with the natural partial order $preceq$. Let $(Z(v))_{vin mathcal{T}}$ be a sequence of independent standard Gaussian random variables and let $alpha = (alpha_k)_{k=1}^infty$ be a sequence of real numbers wit
h $sum_{k=1}^infty alpha_k^2<infty$. Set $alpha_0 =0$ and define a Gaussian process on $mathcal{T}$ in the following way: [ G(mathcal{T}, alpha; v): = sum_{upreceq v} alpha_{|u|} Z(u), quad v in mathcal{T}, ] where $|u|$ denotes the graph distance between the vertex $u$ and the root vertex. Under mild assumptions on $mathcal{T}$, we obtain a necessary and sufficient condition for the almost sure boundedness of the above Gaussian process. Our condition is also necessary and sufficient for the almost sure uniform convergence of the Gaussian process $G(mathcal{T}, alpha; v)$ along all rooted geodesic rays in $mathcal{T}$.
The Patterson-Sullivan construction is proved almost surely to recover a Bergman function from its values on a random discrete subset sampled with the determinantal point process induced by the Bergman kernel on the unit ball $mathbb{D}_d$ in $mathbb
{C}^d$. For super-critical weighted Bergman spaces, the interpolation is uniform when the functions range over the unit ball of the weighted Bergman space. As main results, we obtain a necessary and sufficient condition for interpolation of a fixed pluriharmonic function in the complex hyperbolic space of arbitrary dimension (cf. Theorem 1.4 and Theorem 4.11); optimal simultaneous uniform interpolation for weighted Bergman spaces (cf. Theorem 1.8, Proposition 1.9 and Theorem 4.13); strong simultaneous uniform interpolation for weighted harmonic Hardy spaces (cf. Theorem 1.11 and Theorem 4.15); and establish the impossibility of the uniform simultaneous interpolation for the Bergman space $A^2(mathbb{D}_d)$ on $mathbb{D}_d$ (cf. Theorem 1.12 and Theorem 6.7).
We show that the central limit theorem for linear statistics over determinantal point processes with $J$-Hermitian kernels holds under fairly general conditions. In particular, We establish Gaussian limit for linear statistics over determinantal poin
t processes on union of two copies of $mathbb{R}^d$ when the correlation kernels are $J$-Hermitian translation-invariant.
For a Pfaffian point process we show that its Palm measures, its normalised compositions with multiplicative functionals, and its conditional measures with respect to fixing the configuration in a bounded subset are Pfaffian point processes whose kernels we find explicitly.
The results of this paper are 3-folded. Firstly, for any stationary determinantal process on the integer lattice, induced by strictly positive and strictly contractive involution kernel, we obtain the necessary and sufficient condition for the $psi$-
mixing property. Secondly, we obtain the existence of the $L^q$-dimensions of the stationary determinantal measure on symbolic space ${0, 1}^mathbb{N}$ under appropriate conditions. Thirdly, the previous two results together imply the precise increasing rate of the longest common substring of a typical pair of points in ${0, 1}^mathbb{N}$.
Our first result states that the orthogonal and symplectic Bessel processes are rigid in the sense of Ghosh and Peres. Our argument in the Bessel case proceeds by an estimate of the variance of additive statistics in the spirit of Ghosh and Peres. Se
cond, a sufficient condition for number rigidity of stationary Pfaffian processes, relying on the Kolmogorov criterion for interpolation of stationary processes and applicable, in particular, to pfaffian sine-processes, is given in terms of the asymptotics of the spectral measure for additive statistics.
The Patterson-Sullivan construction is proved almost surely to recover every harmonic function in a certain Banach space from its values on the zero set of a Gaussian analytic function on the disk. The argument relies on the slow growth of variance f
or linear statistics of the concerned point process. As a corollary of reconstruction result in general abstract setting, Patterson-Sullivan reconstruction of harmonic functions is obtained in real and complex hyperbolic spaces of arbitrary dimension.
For a determinantal point process induced by the reproducing kernel of the weighted Bergman space $A^2(U, omega)$ over a domain $U subset mathbb{C}^d$, we establish the mutual absolute continuity of reduced Palm measures of any order provided that th
e domain $U$ contains a non-constant bounded holomorphic function. The result holds in all dimensions. The argument uses the $H^infty(U)$-module structure of $A^2(U, omega)$. A corollary is the quasi-invariance of our determinantal point process under the natural action of the group of compactly supported diffeomorphisms of $U$.
In this note, we give a nature action of the modular group on the ends of the infinite (p + 1)-cayley tree, for each prime p. We show that there is a unique invariant probability measure for each p.
