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Interactions growing slower than a certain exponential of the square of a scalar field, are well behaved when evolved under the functional renormalization group linearised around the Gaussian fixed point. They satisfy properties usually…

We study the expected number of real zeros for random linear combinations of orthogonal polynomials. It is well known that Kac polynomials, spanned by monomials with i.i.d. Gaussian coefficients, have only $(2/\pi + o(1))\log{n}$ expected…

It is shown that the ratios of the quenched averaged three and four-point correlation functions of the local energy density operator to the connected ones in the random-bond Ising model approach asymptotically to some $universal$ functions.…

Correlation functions of primary fields in the Wess-Zumino-Novikov-Witten (WZNW) model are known to satisfy a system of Knizhnik-Zamolodchikov (KZ) equations, which involve constants of motion of the exactly-solvable Gaudin magnet. We…

We evaluate the sum of Gauss hypergeometric functions \[S(\mu,c;x)=\sum_{k\geq 0} \bl(\frac{1-x}{1+\mu}\br)^k\,{}_2F_1(\fs k+\fs, \fs k+1;c;x)\] for $x\in [-1,1]$ and positive parameters $\mu$ and $c$. The domain of absolute convergence of…

The remarkable universality of the eigenvalue correlation functions is perhaps one of the most salient findings in random matrix theory. Particularly for short-range separations of the eigenvalues, the correlation functions have been shown…

We confirm Flandrin's prediction for the expected average of local maxima of spectrograms of complex white noise with Gaussian windows (Gaussian spectrograms or, equivalently, modulus of weighted Gaussian Entire Functions), a consequence of…

In this note we extend the results of the model of galaxy biasing recently proposed in (http://arxiv.org/abs/1212.0868v2). In that paper the authors had outlined a very precise mathematical framework to deal with the theory of galaxy…

We study the zeros and critical points of different indices of the standard Gaussian entire function on the complex plane (whose zero set is stationary). We provide asymptotics for the second order correlations of all the corresponding…

We investigate some issues concerning the zero-momentum four-point renormalized coupling constant g in the symmetric phase of O(N) models, and the corresponding Callan-Symanzik beta-function. In the framework of the 1/N expansion we show…

We study a family of random Taylor series $$F(z) = \sum_{n\ge 0} \zeta_n a_n z^n$$ with radius of convergence almost surely $1$ and independent identically distributed complex Gaussian coefficients $(\zeta_n)$; these Taylor series are…

Two and three point functions of composite operators are analysed with regard to (logarithmically) divergent contact terms. Using the renormalisation group of dimensional regularisation it is established that the divergences are governed by…

Given a function $f(n)$ periodic of period $q\geq 1$ and an irrational number $0<\alpha\leq 1$, Chatterjee and Gun proved that the series $F(s,f,\alpha)=\sum_{n=0}^{\infty}\frac{f(n)}{(n+\alpha)^s}$ has infinitely many zeros for $\sigma>1$…

We compute N-point correlation functions of pure vertex operator states(DK states) for minimal models coupled to gravity. We obtain agreement with the matrix model results on analytically continuing in the numbers of cosmological constant…

We establish a pointwise convergence result for ergodic averages modeled along orbits of the form $(n\lfloor n\sqrt{k}\rfloor)_{n\in\mathbb{N}}$, where $k$ is an arbitrary positive rational number with $\sqrt{k}\not\in\mathbb{Q}$. Namely,…

We discuss structural aspects of the functional renormalisation group. Flows for a general class of correlation functions are derived, and it is shown how symmetry relations of the underlying theory are lifted to the regularised theory. A…

Let $\mathcal S$ denote the class of all functions of the form $f(z)=z+a_2z^2+a_3z^3+\cdots$ which are analytic and univalent in the open unit disk $\ID$ and, for $\lambda >0$, let $\Phi_\lambda (n,f)=\lambda a_n^2-a_{2n-1}$ denote the…

Let $X_{1},X_{2},...$ be a sequence of independent copies (s.i.c) of a real random variable (r.v.) $X\geq 1$, with distribution function $df$ $F(x)=\mathbb{P}% (X\leq x)$ and let $X_{1,n}\leq X_{2,n} \leq ... \leq X_{n,n}$ be the order…

Given any non-polynomial $G$-function $F(z)=\sum\_{k=0}^\infty A\_k z^k$ of radius of convergence $R$, we consider the $G$-functions $F\_n^{[s]}(z)=\sum\_{k=0}^\infty \frac{A\_k}{(k+n)^s}z^k$ for any integers $s\geq 0$ and $n\geq 1$. For…

Let $\{X(t):t\in\mathbb R_+\}$ be a stationary Gaussian process with almost surely (a.s.) continuous sample paths, $\mathbb E X(t) = 0$, $\mathbb E X^2(t) = 1$ and correlation function satisfying (i) $r(t) = 1 - C|t|^{\alpha} +…