arXiv subjects

Jan Nagel

Publications and source records attributed to Jan Nagel.

At least 19 recordsLinked to original sources

arXiv2025

Sum rules via large deviations: polynomial potentials and multi-cut regime on the unit circle

Sum rules are elegant formulas that relate entropy functionals to coefficients associated with orthogonal polynomials [Sim11]. In a series of paper (see for example [GNR16], [GNR17], [BSZ18a], [BSZ18b]), interesting connections have been established between the large theory of spectral measures built on random matrices and sum rules. In this work, we extend this approach by studying sum rules within random matrix models with polynomial potentials on the unit circle, with a particular focus on cases where the equilibrium measure lacks full support.

math.PR↗

arXiv2025

Nonstandard Large and Moderate Deviations for the Laguerre Ensemble

In this paper, we show limit theorems for the weighted spectral measure of the Laguerre ensemble under a nonstandard scaling, when the parameter grows faster than the matrix size. For this parameter scaling, the limit behavior is similar to the case of the Gaussian ensemble. We show a large deviation principle, moderate deviations and a CLT for the spectral measure. For the moderate deviations and the CLT, we observe a particular dependence on the rate of the parameter and a corrective shift by a signed measure. The proofs are based on the tridiagonal representation of the Laguerre ensemble.

math.PR↗

arXiv2024

Central limit theorem for a random walk on Galton-Watson trees with random conductances

We show a central limit theorem for random walk on a Galton-Watson tree, when the edges of the tree are assigned randomly uniformly elliptic conductances. When a positive fraction of edges is assigned a small conductance $\varepsilon$, we study the behavior of the limiting variance as $\varepsilon\to 0$. Provided that the tree formed by larger conductances is supercritical, the variance is nonvanishing as $\varepsilon\to 0$, which implies that the slowdown induced by the $\varepsilon$-edges is not too strong. The proof utilizes a specific regeneration structure, which leads to escape estimates uniform in $\varepsilon$.

math.PR↗

arXiv2023

A new paradigm for the efficient inclusion of stochasticity in engineering simulations

As a physical fact, randomness is an inherent and ineliminable aspect in all physical measurements and engineering production. As a consequence, material parameters, serving as input data, are only known in a stochastic sense and thus, also output parameters, e.g., stresses, fluctuate. For the estimation of those fluctuations it is imperative to incoporate randomness into engineering simulations. Unfortunately, incorporating uncertain parameters into the modeling and simulation of inelastic materials is often computationally expensive, as many individual simulations may have to be performed. The promise of the proposed method is simple: using extended material models to include stochasticity reduces the number of needed simulations to one. This single computation is cheap, i.e., it has a comparable numerical effort as a single standard simulation. The extended material models are easily derived from standard deterministic material models and account for the effect of uncertainty by an extended set of deterministic material parameters. The time-dependent and stochastic material behavior are separated, such that only the deterministic time-dependent behavior of the extended material model needs to be simulated. The effect of stochasticity is then included during post-processing. The feasibility of this approach is demonstrated for three different and highly non-linear material models: viscous damage, viscous phase transformations and elasto-viscoplasticity. A comparison to the Monte Carlo method showcases that the method is indeed able to provide reliable estimates of the expectation and variance of internal variables and stress at a minimal fraction of the computation cost.

cs.CE↗

arXiv2023

Cohomology of complete intersections of quadrics

We show that the variable cohomology of a general complete intersection of quadrics can be identified with the intersection cohomology of a double covering. As a consequence, we show that the middle cohomology of a general complete intersection of four quadrics in an odd-dimensional projective space is isomorphic to the middle cohomology of a resolution of singularities of a double solid.

math.AG↗

arXiv2023

On some gateways between sum rules

We present correspondences induced by some classical mappings between measures on an interval and measures on the unit circle. More precisely, we link their sequences of orthogonal polynomial and their recursion coefficients. We also deduce some correspondences between particular equilibrium measures of random matrix ensembles. Additionally, we show that these mappings open up gateways between the sum rules associated with some classical models, leading to new formulations of several sum rules.

math.PR↗

arXiv2022

Motivic decompositions of families with Tate fibers: smooth and singular cases

We apply Wildeshaus's theory of motivic intermediate extensions to the motivic decomposition conjecture, formulated by Deninger-Murre and Corti-Hanamura. We first obtain a general motivic decomposition for the Chow motive of an arbitrary smooth projective family $f:X \rightarrow S$ whose geometric fibers are Tate. Using Voevodsky's motives with rational coefficients, the formula is valid for an arbitrary regular base $S$, without assuming the existence of a base field or even of a prime integer $\ell$ invertible on $S$. This result, and some of Bondarko' ideas, lead us to a generalized formulation of Corti-Hanamura's conjecture. Secondly we establish the existence of the motivic decomposition when $f:X \rightarrow S$ is a projective quadric bundle over a characteristic $0$ base, which is either sufficiently general or whose discriminant locus is a normal crossing divisor. This provides a motivic lift of the Bernstein-Beilinson-Deligne decomposition in this setting.

math.AG↗

arXiv2020

The speed of random walk on Galton-Watson trees with vanishing conductances

In this paper we consider random walks on Galton-Watson trees with random conductances. On these trees, the distance of the walker to the root satisfies a law of large numbers with limit the effective velocity, or speed of the walk. We study the regularity of the speed as a function of the distribution of conductances, in particular when the distribution of conductances converges to a non-elliptic limit.

math.PR↗

arXiv2020

Sum rules via large deviations: extension to polynomial potentials and the multi-cut regime

A sum rule is an identity connecting the entropy of a measure with coefficients involved in the construction of its orthogonal polynomials (Jacobi coefficients). Our paper is an extension of Gamboa, Nagel and Rouault (2016), where we have showed sum rules by using only probabilistic tools (namely the large deviations theory). Here, we prove large deviation principles for the weighted spectral measure of unitarily invariant random matrices in two general situations: firstly, when the equilibrium measure is not necessarily supported by a single interval and secondly, when the potential is a nonnegative polynomial. The rate functions can be expressed as functions of the Jacobi coefficients. These new large deviation results lead to original sum rules both for the one and the multi-cut regime and also answer a conjecture stated in Gamboa, Nagel and Rouault (2016) concerning general sum rules.

math.PR↗

arXiv2019

A functional CLT for partial traces of random matrices

In this paper we show a functional central limit theorem for the sum of the first $\lfloor t n \rfloor$ diagonal elements of $f(Z)$ as a function in $t$, for $Z$ a random real symmetric or complex Hermitian $n\times n$ matrix. The result holds for orthogonal or unitarily invariant distributions of $Z$, in the cases when the linear eigenvalue statistic $\operatorname{tr} f(Z)$ satisfies a CLT. The limit process interpolates between the fluctuations of individual matrix elements as $f(Z)_{1,1}$ and of the linear eigenvalue statistic. It can also be seen as a functional CLT for processes of randomly weighted measures.

math.PR↗

arXiv2019

The homotopy Leray spectral sequence

In this work, we build a spectral sequence in motivic homotopy that is analogous to both the Serre spectral sequence in algebraic topology and the Leray spectral sequence in algebraic geometry. Here, we focus on laying the foundations necessary to build the spectral sequence and give a convenient description of its $E_2$-page. Our description of the $E_2$-page is in terms of homology of the local system of fibers, which is given using a theory similar to Rost's cycle modules. We close by providing some sample applications of the spectral sequence and some hints at future work.

math.AG↗

arXiv2019

Random walk on barely supercritical branching random walk

Let $\mathcal{T}$ be a supercritical Galton-Watson tree with a bounded offspring distribution that has mean $μ>1$, conditioned to survive. Let $φ_{\mathcal{T}}$ be a random embedding of $\mathcal{T}$ into $\mathbb{Z}^d$ according to a simple random walk step distribution. Let $\mathcal{T}_p$ be percolation on $\mathcal{T}$ with parameter $p$, and let $p_c = μ^{-1}$ be the critical percolation parameter. We consider a random walk $(X_n)_{n \ge 1}$ on $\mathcal{T}_p$ and investigate the behavior of the embedded process $φ_{\mathcal{T}_p}(X_n)$ as $n\to \infty$ and simultaneously, $\mathcal{T}_p$ becomes critical, that is, $p=p_n\searrow p_c$. We show that when we scale time by $n/(p_n-p_c)^3$ and space by $\sqrt{(p_n-p_c)/n}$, the process $(φ_{\mathcal{T}_p}(X_n))_{n \ge 1}$ converges to a $d$-dimensional Brownian motion. We argue that this scaling can be seen as an interpolation between the scaling of random walk on a static random tree and the anomalous scaling of processes in critical random environments.

math.PR↗

arXiv2018

Sum rules and large deviations for spectral matrix measures in the Jacobi ensemble

We continue to explore the connections between large deviations for objects coming from random matrix theory and sum rules. This connection was established in [17] for spectral measures of classical ensembles (Gauss-Hermite, Laguerre, Jacobi) and it was extended to spectral matrix measures of the Hermite and Laguerre ensemble in [20]. In this paper, we consider the remaining case of spectral matrix measures of the Jacobi ensemble. Our main results are a large deviation principle for such measures and a sum rule for matrix measures with reference measure the Kesten-McKay law. As an important intermediate step, we derive the distribution of canonical moments of the matrix Jacobi ensemble.

math.PR↗

arXiv2017

On complete intersections in varieties with finite-dimensional motive

Let $X$ be a complete intersection inside a variety $M$ with finite dimensional motive and for which the Lefschetz-type conjecture $B(M)$ holds. We show how conditions on the niveau filtration on the homology of $X$ influence directly the niveau on the level of Chow groups. This leads to a generalization of Voisin's result. The latter states that if $M$ has trivial Chow groups and if $X$ has non-trivial variable cohomology parametrized by $c$-dimensional algebraic cycles, then the cycle class maps $A_k(X) \to H_{2k}(X)$ are injective for $k<c$. We give variants involving group actions which lead to several new examples with finite dimensional Chow motives.

math.AG↗

arXiv2017

The speed of biased random walk among random conductances

We consider biased random walk among iid, uniformly elliptic conductances on $\mathbb{Z}^d$, and investigate the monotonicity of the velocity as a function of the bias. It is not hard to see that if the bias is large enough, the velocity is increasing as a function of the bias. Our main result is that if the disorder is small, i.e. all the conductances are close enough to each other, the velocity is always strictly increasing as a function of the bias, see Theorem 1. A crucial ingredient of the proof is a formula for the derivative of the velocity, which can be written as a covariance, see Theorem 3: it follows along the lines of the proof of the Einstein relation in [GGN]. On the other hand, we give a counterexample showing that for iid, uniformly elliptic conductances, the velocity is not always increasing as a function of the bias. More precisely, if $d=2$ and if the conductances take the values $1$ (with probability $p$) and $κ$ (with probability $1-p$) and $p$ is close enough to $1$ and $κ$ small enough, the velocity is not increasing as a function of the bias, see Theorem 2.

math.PR↗

arXiv2017

Addendum to " Sum rules via large deviations "

In these notes we fill a gap in a proof in Section 4 of Gamboa, Nagel, Rouault [Sum rules via large deviations, J. Funct. Anal. 270 (2016), 509-559]. We prove a general theorem which combines a LDP with a convex rate function and a LDP with a non-convex one. This result will be used to prove LDPs for spectral matrix measures and for spectral measures on the unit circle.

math.PR↗

arXiv2017

Sum rules and large deviations for spectral measures on the unit circle

This work is a companion paper of Gamboa, Nagel, Rouault (J. Funct. Anal. 2016). We continue to explore the connections between large deviations for random objects issued from random matrix theory and sum rules. Here, we are concerned essentially with measures on the unit circle whose support is an arc that is possibly proper. We particularly focus on two matrix models. The first one is the Gross-Witten ensemble. In the gapped regime we give a probabilistic interpretation of a Simon sum rule. The second matrix model is the Hua-Pickrell ensemble. Unlike the Gross-Witten ensemble the potential is here infinite at one point. Surprisingly, but as in the above mentioned paper, we obtain a completely new sum rule for the deviation to the equilibrium measure of the Hua-Pickrell ensemble. The extension to matrix measures is also studied.

math.PR↗

arXiv2016

Sum rules and large deviations for spectral matrix measures

A sum rule relative to a reference measure on R is a relationship between the reversed Kullback-Leibler divergence of a positive measure on R and some non-linear functional built on spectral elements related to this measure (see for example Killip and Simon 2003). In this paper, using only probabilistic tools of large deviations, we extend the sum rules obtained in Gamboa, Nagel and Rouault (2015) to the case of Hermitian matrix-valued measures. We recover the earlier result of Damanik, Killip and Simon (2010) when the reference measure is the (matrix-valued) semicircle law and obtain a new sum rule when the reference measure is the (matrix-valued) Marchenko-Pastur law.

math.PR↗