PDE's for the Dyson, Airy and Sine processes

Mark Adler

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Displaying similar documents to “PDE's for the Dyson, Airy and Sine processes”

The ratio and generating function of cogrowth coefficients of finitely generated groups

Ryszard Szwarc (1998)

Studia Mathematica

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Let G be a group generated by r elements $g_{1}, \dots, g_{r}$ . Among the reduced words in $g_{1}, \dots, g_{r}$ of length n some, say $γ_{n}$ , represent the identity element of the group G. It has been shown in a combinatorial way that the 2nth root of $γ_{2 n}$ has a limit, called the cogrowth exponent with respect to the generators $g_{1}, \dots, g_{r}$ . We show by analytic methods that the numbers $γ_{n}$ vary regularly, i.e. the ratio $γ_{2 n + 2} / γ_{2 n}$ is also convergent. Moreover, we derive new precise information on the domain of holomorphy of γ(z), the generating function...

Non-vanishing of class group $L$ -functions at the central point

Valentin Blomer (2004)

Annales de l’institut Fourier

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Let $K = ℚ (\sqrt{- D})$ be an imaginary quadratic field, and denote by $h$ its class number. It is shown that there is an absolute constant $c > 0$ such that for sufficiently large $D$ at least $c \cdot h \prod_{p ∣ D} (1 - p^{- 1})$ of the $h$ distinct $L$ -functions $L_{K} (s, χ)$ do not vanish at the central point $s = 1 / 2$ .

On the exponential Orlicz norms of stopped Brownian motion

Goran Peškir (1996)

Studia Mathematica

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Necessary and sufficient conditions are found for the exponential Orlicz norm (generated by $ψ_{p} (x) = {e x p (| x |}^{p}) - 1$ with 0 < p ≤ 2) of $m a x_{0 \leq t \leq τ} | B_{t} |$ or $| B_{τ} |$ to be finite, where $B = {(B_{t})}_{t \geq 0}$ is a standard Brownian motion and τ is a stopping time for B. The conditions are in terms of the moments of the stopping time τ. For instance, we find that $∥ m a x_{0 \leq t \leq τ} | B_{t} | ∥_{ψ_{1}} < \infty$ as soon as $E (τ^{k}) = O (C^{k} k^{k})$ for some constant C > 0 as k → ∞ (or equivalently $∥ τ ∥_{ψ_{1}} < \infty$ ). In particular, if τ ∼ Exp(λ) or $| N (0, σ^{2}) |$ then the last condition is satisfied, and we obtain $∥ m a x_{0 \leq t \leq τ} | B_{t} | ∥_{ψ_{1}} \leq K \sqrt E (τ)$ with some universal constant...

On maximum increase and decrease of brownian motion

Paavo Salminen, Pierre Vallois (2007)

Annales de l'I.H.P. Probabilités et statistiques

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On linear normal lattices configurations

Mordechay B. Levin, Meir Smorodinsky (2005)

Journal de Théorie des Nombres de Bordeaux

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In this paper we extend Champernowne’s construction of normal numbers in base $b$ to the $ℤ^{d}$ case and obtain an explicit construction of the generic point of the $ℤ^{d}$ shift transformation of the set ${0, 1, . . ., b - 1}^{ℤ^{d}}$ . We prove that the intersection of the considered lattice configuration with an arbitrary line is a normal sequence in base $b$ .

Rates of convergence to the local time of a diffusion

Jean Jacod (1998)

Annales de l'I.H.P. Probabilités et statistiques

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On the Maximum of a Branching Process Conditioned on the Total Progeny

Kerbashev, Tzvetozar (1999)

Serdica Mathematical Journal

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The maximum M of a critical Bienaymé-Galton-Watson process conditioned on the total progeny N is studied. Imbedding of the process in a random walk is used. A limit theorem for the distribution of M as N → ∞ is proved. The result is trasferred to the non-critical processes. A corollary for the maximal strata of a random rooted labeled tree is obtained.

Displaying similar documents to “PDE's for the Dyson, Airy and Sine processes”

The ratio and generating function of cogrowth coefficients of finitely generated groups

Non-vanishing of class group L -functions at the central point

On the exponential Orlicz norms of stopped Brownian motion

On maximum increase and decrease of brownian motion

On linear normal lattices configurations

Rates of convergence to the local time of a diffusion

On the Maximum of a Branching Process Conditioned on the Total Progeny

Non-vanishing of class group $L$ -functions at the central point