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Joseph Najnudel

Publications and source records attributed to Joseph Najnudel.

At least 55 records · Page 3Linked to original sources
arXiv2017

On the maximum of the C$β$E field

In this paper, we investigate the extremal values of (the logarithm of) the characteristic polynomial of a random unitary matrix whose spectrum is distributed according the Circular Beta Ensemble (C$β$E). More precisely, if $X_n$ is this characteristic polynomial and $\mathbb{U}$ the unit circle, we prove that: $$\sup_{z \in \mathbb{U} } \Re \log X_n(z) = \sqrt{\frac{2}β} \left(\log n - \frac{3}{4} \log \log n + \mathcal{O}(1) \right)\ ,$$ as well as an analogous statement for the imaginary part. The notation $\mathcal{O}(1)$ means that the corresponding family of random variables, indexed by $n$, is tight. This answers a conjecture of Fyodorov, Hiary and Keating, originally formulated for the case where $β$ equals to $2$, which corresponds to the CUE field.

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arXiv2015

The Circular Unitary Ensemble and the Riemann zeta function: the microscopic landscape and a new approach to ratios

We show in this paper that after proper scalings, the characteristic polynomial of a random unitary matrix converges almost surely to a random analytic function whose zeros, which are on the real line, form a determinantal point process with sine kernel. Our scaling is performed at the so-called "microscopic" level, that is we consider the characteristic polynomial at points which are of order $1/n$ distant. We prove this in the framework of virtual isometries to circumvent the fact that the rescaled characteristic polynomial does not even have a moment of order one, hence making the classical techniques of random matrix theory difficult to apply. The strong convergence results in this setup provide us with a new approach to ratios: we are able to solve open problems about the limiting distribution of ratios of characteristic polynomials evaluated at points of the form $\exp(2 i πα/n)$ and related objects (such as the logarithmic derivative). We also explicitly describe the dependence relation for the logarithm of the characteristic polynomial evaluated at several points on the microscopic scale. On the number theory side, inspired by the Keating-Snaith philosophy, we conjecture some new limit theorems for the value distribution of the Riemann zeta function on the critical line at the stochastic process level.

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arXiv2014

On sigma-finite measures related to the Martin boundary of recurrent Markov chains

In our monograph with B. Roynette and M. Yor, we construct a sigma-finite measure related to penalisations of different stochastic processes, including the Brownian motion in dimension 1 or 2, and a large class of linear diffusions. In the last chapter of the monograph, we define similar measures from recurrent Markov chains satisfying some technical conditions. In the present paper, we give a classification of these measures, in function of the minimal Martin boundary of the Markov chain considered at the beginning. We apply this classification to the examples considered at the end of our monograph.

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arXiv2014

Limit operators for circular ensembles

It is known that a unitary matrix can be decomposed into a product of reflections, one for each dimension, and the Haar measure on the unitary group pushes forward to independent uniform measures on the reflections. We consider the sequence of unitary matrices given by successive products of random reflections. In this coupling, we show that powers of the sequence of matrices converge in a suitable sense to a flow of operators which acts on a random vector space. The vector space has an explicit description as a subspace of the space of sequences of complex numbers. The eigenvalues of the matrices converge almost surely to the eigenvalues of the flow, which are distributed in law according to a sine-kernel point process. The eigenvectors of the matrices converge almost surely to vectors which are distributed in law as Gaussian random fields on a countable set. This flow gives the first example of a random operator with a spectrum distributed according to a sine-kernel point process which is naturally constructed from finite dimensional random matrix theory.

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arXiv2014

A limiting random analytic function related to the CUE

We show in this paper that, when properly rescaled in time and in space, the characteristic polynomial of a random unitary matrix converges almost surely to a random analytic function whose zeros, which are on the real line, form a determinantal point process with sine kernel. We prove this result in the framework of virtual isometries to circumvent the fact that the rescaled characteristic polynomial does not even have a moment of order one, hence making the classical techniques of random matrix theory difficult to apply.

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arXiv2013

Random permutation matrices under the generalized Ewens measure

We consider a generalization of the Ewens measure for the symmetric group, calculating moments of the characteristic polynomial and similar multiplicative statistics. In addition, we study the asymptotic behavior of linear statistics (such as the trace of a permutation matrix or of a wreath product) under this new measure.

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arXiv2013

Limit theorems for orthogonal polynomials related to circular ensembles

For a natural extension of the circular unitary ensemble of order n, we study as n tends to infinity, the asymptotic behavior of the sequence of orthogonal polynomials with respect to the spectral measure. The last term of this sequence is the characteristic polynomial. After taking logarithm and rescaling, we obtain a process indexed by t in [0,1]. We show that it converges to a deterministic limit, and we describe the fluctuations and the large deviations.

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arXiv2013

On the number of zeros of linear combinations of independent characteristic polynomials of random unitary matrices

We show that almost all the zeros of any finite linear combination of independent characteristic polynomials of random unitary matrices lie on the unit circle. This result is the random matrix analogue of an earlier result by Bombieri and Hejhal on the distribution of zeros of linear combinations of $L$-functions, thus providing further evidence for the conjectured links between the value distribution of the characteristic polynomial of random unitary matrices and the value distribution of $L$-functions on the critical line.

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arXiv2011

A new kind of augmentation of filtrations

Let $(Ω,\mathcal{F},(\mathcal{F}_t)_{t \geq 0},\mathbb{P})$ be a filtered probability space satisfying the usual assumptions: it is usually not possible to extend to $\mathcal{F}_{\infty}$ (the $σ$-algebra generated by $(\mathcal{F}_t)_{t \geq 0}$) a coherent family of probability measures $(\mathbb{Q}_t)$, each of them being defined on $\mathcal{F}_t$. It is known that for instance, on the Wiener space, this extension problem has a positive answer if one takes the filtration generated by the coordinate process, but can have a negative answer if one takes its usual augmentation. On the other hand, the usual assumptions are crucial in order to obtain the existence of regular versions of paths for most stochastic processes of interest, such as the local time of the standard Brownian motion, stochastic integrals, etc. In order to fix this problem, we introduce a new property for filtrations, intermediate between the right continuity and the usual conditions. We show that most of the important results of the theory of stochastic processes which are generally proved under the usual augmentation, such as the existence of regular version of trajectories or the début theorem, still hold under the N-augmentation; moreover this new augmentation allows the extension of a coherent family of probability measures whenever this is possible with the original filtration.

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arXiv2010

Construction of an Edwards' probability measure on $\mathcal{C}(\mathbb{R}_+,\mathbb{R})$

In this article, we prove that the measures $\mathbb{Q}_T$ associated to the one-dimensional Edwards' model on the interval $[0,T]$ converge to a limit measure $\mathbb{Q}$ when $T$ goes to infinity, in the following sense: for all $s\geq0$ and for all events $Λ_s$ depending on the canonical process only up to time $s$, $\mathbb{Q}_T(Λ_s)\rightarrow\mathbb{Q}(Λ_s)$. Moreover, we prove that, if $\mathbb{P}$ is Wiener measure, there exists a martingale $(D_s)_{s\in\mathbb{R}_+}$ such that $\mathbb{Q}(Λ_s) =\mathbb{E}_{\mathbb{P}}(\mathbh{1}_{Λ_s}D_s)$, and we give an explicit expression for this martingale.

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arXiv2010

On a flow of operators associated to virtual permutations

Kerov, Olshanski and Vershik introduced the so-called virtual permutations, defined as families of permutations $(σ_N)_{N \geq 1}$, $σ_N$ in the symmetric group of order $N$, such that the cycle structure of $σ_N$ can be deduced from the structure of $σ_{N+1}$ simply by removing the element $N+1$. The virtual permutations, and in particular the probability measures on the corresponding space which are invariant by conjugation, have been studied in a details by Tsilevich. In the present article, we prove that for a large class of such invariant measures (containing in particular the Ewens measure of any parameter $θ\geq 0$), it is possible to associate a flow $(T^α)_{α\in \mathbb{R}}$ of random operators on a suitable functional space. Moreover, if $(σ_N)_{N \geq 1}$ is a random virtual permutation following a distribution in the class described above, the operator $T^α$ can be interpreted as the limit, in a sense which has to be made precise, of the permutation $σ_N^{α_N}$, where $N$ goes to infinity and $α_N$ is equivalent to $αN$. In relation with this interpretation, we prove that the eigenvalues of the infinitesimal generator of $(T^α)_{α\in \mathbb{R}}$ are equal to the limit of the rescaled eigenangles of the permutation matrix associated to $σ_N$.

math.PR↗
arXiv2010

The distribution of eigenvalues of randomized permutation matrices

In this article we study in detail a family of random matrix ensembles which are obtained from random permutations matrices (chosen at random according to the Ewens measure of parameter $θ>0$) by replacing the entries equal to one by more general non-vanishing complex random variables. For these ensembles, in contrast with more classical models as the Gaussian Unitary Ensemble, or the Circular Unitary Ensemble, the eigenvalues can be very explicitly computed by using the cycle structure of the permutations. Moreover, by using the so-called virtual permutations, first introduced by Kerov, Olshanski and Vershik, and studied with a probabilistic point of view by Tsilevich, we are able to define, on the same probability space, a model for each dimension greater than or equal to one, which gives a meaning to the notion of almost sure convergence when the dimension tends to infinity. In the present paper, depending on the precise model which is considered, we obtain a number of different results of convergence for the point measure of the eigenvalues, some of these results giving a strong convergence, which is not common in random matrix theory.

math.PR↗
arXiv2009

A new construction of the $σ$-finite measures associated with submartingales of class $(Σ)$

In a previous paper, we proved that for any submartingale $(X_t)_{t \geq 0}$ of class $(Σ)$, defined on a filtered probability space $(Ω, \mathcal{F}, \mathbb{P}, (\mathcal{F}_t)_{t \geq 0})$, which satisfies some technical conditions, one can construct a $σ$-finite measure $\mathcal{Q}$ on $(Ω, \mathcal{F})$, such that for all $t \geq 0$, and for all events $Λ_t \in \mathcal{F}_t$: $$ \mathcal{Q} [Λ_t, g\leq t] = \mathbb{E}_{\mathbb{P}} [\mathds{1}_{Λ_t} X_t]$$ where $g$ is the last hitting time of zero of the process $X$. Some particular cases of this construction are related with Brownian penalisation or mathematical finance. In this note, we give a simpler construction of $\mathcal{Q}$, and we show that an analog of this measure can also be defined for discrete-time submartingales.

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arXiv2009

On penalisation results related with a remarkable class of submartingales

Is this paper we study penalisations of diffusions satisfying some technical conditions, generalizing a result obtained by Najnudel, Roynette and Yor. If one of these diffusions has probability distribution $\mathbb{P}$, then our result can be described as follows: for a large class of families of probability measures $(\mathbb{Q}_t)_{t \geq 0}$, each of them being absolutely continuous with respect to $\mathbb{P}$, there exists a probability $\mathbb{Q}_{\infty}$ such that for all events $Λ$ depending only on the canonical trajectory up to a fixed time, $\mathbb{Q}_t (Λ)$ tends to $\mathbb{Q}_{\infty} (Λ)$ when $t$ goes to infinity. In the cases we study here, the limit measure $\mathbb{Q}_{\infty}$ is absolutely continous with respect to a sigma-finite measure $\mathcal{Q}$, which does not depend on the choice of the family of probabilities $(\mathbb{Q}_t)_{t \geq 0}$, but only on $\mathbb{P}$. The relation between $\mathbb{P}$ and $\mathcal{Q}$ is obtained in a very general framework by the authors of this paper.

math.PR↗
arXiv2009

On some properties of a universal sigma-finite measure associated with a remarkable class of submartingales

In a previous work, we associated with any submartingale $X$ of class $(Σ)$, defined on a filtered probability space $(Ω, \mathcal{F}, \mathbb{P}, (\mathcal{F}_t)_{t \geq 0})$ satisfying some technical conditions, a $σ$-finite measure $\mathcal{Q}$ on $(Ω, \mathcal{F})$, such that for all $t \geq 0$, and for all events $Λ_t \in \mathcal{F}_t$: $$ \mathcal{Q} [Λ_t, g\leq t] = \mathbb{E}_{\mathbb{P}} [\mathds{1}_{Λ_t} X_t],$$ where $g$ is the last hitting time of zero by the process $X$. The measure $\mathcal{Q}$, which was previously studied in particular cases related with Brownian penalisations and problems in mathematical finance, enjoys some remarkable properties which are detailed in this paper. Most of these properties are related to a certain class of nonnegative martingales, defined as the local densities (with respect to $\mathbb{P}$) of the finite measures which are absolutely continuous with respect to $\mathcal{Q}$. From the properties of the measure $\mathcal{Q}$, we also deduce a universal class of penalisation results of the probability measure $\mathbb{P}$ with a large class of functionals: the measure $\mathcal{Q}$ appears to be the unifying object in these problems.

math.PR↗
arXiv2009

Scaled limit and rate of convergence for the largest eigenvalue from the generalized Cauchy random matrix ensemble

In this paper, we are interested in the asymptotic properties for the largest eigenvalue of the Hermitian random matrix ensemble, called the Generalized Cauchy ensemble $GCy$, whose eigenvalues PDF is given by $$\textrm{const}\cdot\prod_{1\leq j -1/2$ and where $N$ is the size of the matrix ensemble. Using results by Borodin and Olshanski \cite{Borodin-Olshanski}, we first prove that for this ensemble, the largest eigenvalue divided by $N$ converges in law to some probability distribution for all $s$ such that $\Re(s)>-1/2$. Using results by Forrester and Witte \cite{Forrester-Witte2} on the distribution of the largest eigenvalue for fixed $N$, we also express the limiting probability distribution in terms of some non-linear second order differential equation. Eventually, we show that the convergence of the probability distribution function of the re-scaled largest eigenvalue to the limiting one is at least of order $(1/N)$.

math.PR↗
arXiv2009

On some universal sigma finite measures and some extensions of Doob's optional stopping theorem

In this paper, we associate, to any submartingale of class $(Σ)$, defined on a filtered probability space $(Ω, \mathcal{F}, \mathbb{P}, (\mathcal{F}_t)_{t \geq 0})$, which satisfies some technical conditions, a $σ$-finite measure $\mathcal{Q}$ on $(Ω, \mathcal{F})$, such that for all $t \geq 0$, and for all events $Λ_t \in \mathcal{F}_t$: $$ \mathcal{Q} [Λ_t, g\leq t] = \mathbb{E}_{\mathbb{P}} [\mathds{1}_{Λ_t} X_t]$$ where $g$ is the last hitting time of zero of the process $X$. This measure $\mathcal{Q}$ has already been defined in several particular cases, some of them are involved in the study of Brownian penalisation, and others are related with problems in mathematical finance. More precisely, the existence of $\mathcal{Q}$ in the general case solves a problem stated by D. Madan, B. Roynette and M. Yor, in a paper studying the link between Black-Scholes formula and last passage times of certain submartingales. Moreover, the equality defining $\mathcal{Q}$ remains true if one replaces the fixed time $t$ by any bounded stopping time. This generalization can be viewed as an extension of Doob's optional stopping theorem.

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