We study the typical profiles of a one dimensional random field Kac model, for values of the temperature and magnitude of the field in the region of the two absolute minima for the free energy of the corresponding random field Curie Weiss model. We show that, for a set of realizations of the random field of overwhelming probability, the localization of the two phases corresponding to the previous minima is completely determined. Namely, we are able to construct random intervals tagged with a sign, where typically, with respect to the infinite volume Gibbs measure, the profile is rigid and takes, according to the sign, one of the two values corresponding to the previous minima. Moreover, we characterize the transition from one phase to the other.
We study a spin-flip model with Kac type interaction, in the presence of a random field given by i.i.d. bounded random variables. The system, spatially inhomogeneous, evolves according to a non conservative (Glauber) dynamics. We show an almost sure (with respect to the random field) large deviations principle for the empirical magnetizations of this process. The rate functional depends on the statistical properties of the external random field, it is lower semicontinuous with compact level sets.
In this article we prove existence of the asymptotic entropy for isotropic random walks on regular Fuchsian buildings. Moreover, we give formulae for the asymptotic entropy, and prove that it is equal to the rate of escape of the random walk with respect to the Green distance. When the building arises from a Fuchsian Kac-Moody group our results imply results for random walks induced by bi-invariant measures on these groups, however our results are proven in the general setting without the assumption of any group acting on the building. The main idea is to consider the retraction of the isotrop…
We show that the classical Kac's random walk on $(n-1)$-sphere $S^{n-1}$ starting from the point mass at $e_1$ mixes in $\mathcal{O}(n^5(\log n)^3)$ steps in total variation distance. The main argument uses a truncation of the running density after a burn-in period, followed by $\mathcal{L}^2$ convergence using the spectral gap information derived by other authors. This improves upon a previous bound by Diaconis and Saloff-Coste of order $\mathcal {O}(n^{2n})$.
We give examples of piecewise linear random interval maps satisfying arcsine and Darling--Kac laws, which are analogous to Thaler's arcsine and Aaronson's Darling--Kac laws for the Boole transform. They are constructed by random switch of two piecewise linear maps with attracting or repelling fixed points, which behave as if they were indifferent fixed points of a deterministic map.
We prove that the largest and smallest root in modulus of random Kac polynomials have a non-universal behavior. They do not converge towards the edge of the support of the limiting distribution of the zeros. This non-universality is surprising as the large deviation principle for the empirical measure is universal. This is in sharp contrast with random matrix theory where the large deviation principle is non-universal but the fluctuations of the largest eigenvalue are universal. We show that the modulus of the largest zero is heavy tailed, with a number of finite moments bounded from above by …
We prove the universality of the large deviations principle for the empirical measures of zeros of random polynomials whose coefficients are i.i.d. random variables possessing a density with respect to the Lebesgue measure on C, R or R + , under the assumption that the density does not vanish too fast at zero and decays at least as exp --|x| $ρ$ , $ρ$ \textgreater{} 0, at infinity.
Estimate (3.39) which appears in the proof of Proposition 3.4 in [Ann. Probab. 27 (1999) 1414--1467, doi:10.1214/aop/1022677454] is wrong. We present below a corrected proof which introduces an extra factor 2 in equations (3.34) and (3.35). This has no consequence in the rest of the paper since Proposition 3.4 is used to estimate only ratios; see (3.23) and (3.25).
We prove a quenched weak large deviations principle for the Gibbs measures of a Random Field Kac Model (RFKM) in one dimension. The external random magnetic field is given by symmetrically distributed Bernoulli random variables. The results are valid for values of the temperature, $β^{-1}$, and magnitude, $θ$, of the field in the region where the free energy of the corresponding random Curie Weiss model has only two absolute minima $m_β$ and $Tm_β$. We give an explicit representation of the rate functional which is a positive random functional determined by two distinct contributions. One is r…
This work develops Feynman-Kac formulas for a class of regime-switching jump diffusion processes, in which the jump part is driven by a Poisson random measure associated to a general Lévy process and the switching part depends on the jump diffusion processes. Under broad conditions, the connections of such stochastic processes and the corresponding partial integro-differential equations are established. Related initial, terminal, and boundary value problems are also treated. Moreover, based on weak convergence of probability measures, it is demonstrated that a sequence of random variables rela…
We investigate the fluctuations and large deviations of the root of largest modulus in a model of random polynomial with independent complex Gaussian coefficients (Kac polynomials). The fluctuations were recently computed by R. Butez (arxiv 1704.02761) and involve a Fredholm determinant. The precise large deviations show a particular function defined by a series of mutiple integrals in the same vein.
We consider Kac's random walk on $n$-dimensional rotation matrices, where each step is a random rotation in the plane generated by two randomly picked coordinates. We show that this process converges to the Haar measure on $\mathit{SO}(n)$ in the $L^2$ transportation cost (Wasserstein) metric in $O(n^2\ln n)$ steps. We also prove that our bound is at most a $O(\ln n)$ factor away from optimal. Previous bounds, due to Diaconis/Saloff-Coste and Pak/Sidenko, had extra powers of $n$ and held only for $L^1$ transportation cost. Our proof method includes a general result of independent interest, aki…
In this paper we prove a rate of escape theorem and a central limit theorem for isotropic random walks on Fuchsian buildings, giving formulae for the speed and asymptotic variance. In particular, these results apply to random walks induced by bi-invariant measures on Fuchsian Kac-Moody groups, however they also apply to the case where the building is not associated to any reasonable group structure. Our primary strategy is to construct a renewal structure of the random walk. For this purpose we define cones and cone types for buildings and prove that the corresponding automata in the building …
Random paths are time continuous interpolations of random walks. By using Littelmann path model, we associate to each irreducible highest weight module of a Kac Moody algebra g a random path W. Under suitable hypotheses, we make explicit the probability of the event E: W never exits the Weyl chamber of g. We then give the law of the random walk defined by W conditioned by the event E and proves this law can be recovered by applying to W the generalized Pitmann transform introduced by Biane, Bougerol and O'Connell. This generalizes the main results of [10] and [16] to Kac Moody root systems and…
In this paper, we study the number of real roots of random trigonometric polynomials with iid coefficients. When the coefficients have zero mean, unit variance and some finite high moments, we show that the variance of the number of real roots is asymptotically linear in terms of the expectation; furthermore, the multiplicative constant in this linear relationship depends only on the kurtosis of the common distribution of the polynomial's coefficients. This result is in sharp contrast to the classical Kac polynomials whose corresponding variance depends only on the first two moments. Our resul…
We develop a technique for proving number rigidity (in the sense of Ghosh-Peres) of the spectrum of general random Schrödinger operators (RSOs). Our method makes use of Feynman-Kac formulas to estimate the variance of exponential linear statistics of the spectrum in terms of self-intersection local times. Inspired by recent results concerning Feynman-Kac formulas for RSOs with multiplicative white noise by Gorin, Shkolnikov and the first-named author, we use this method to prove number rigidity for a class of one-dimensional continuous RSOs of the form $-\frac12Δ+V+ξ$, where $V$ is a determini…
Determining the total variation mixing time of Kac's random walk on the special orthogonal group $\mathrm{SO}(n)$ has been a long-standing open problem. In this paper, we construct a novel non-Markovian coupling for bounding this mixing time. The analysis of our coupling entails controlling the smallest singular value of a certain random matrix with highly dependent entries. The dependence of the entries in our matrix makes it not-amenable to existing techniques in random matrix theory. To circumvent this difficulty, we extend some recent bounds on the smallest singular values of matrices with…
We consider an analogue of the Kac random walk on the special orthogonal group $SO(N)$, in which at each step a random rotation is performed in a randomly chosen 2-plane of $\bR^N$. We obtain sharp asymptotics for the rate of convergence in total variance distance, establishing a cut-off phenomenon in the large $N$ limit. In the special case where the angle of rotation is deterministic this confirms a conjecture of Rosenthal \cite{Rosenthal}. Under mild conditions we also establish a cut-off for convergence of the walk to stationarity under the $L^2$ norm. Depending on the distribution of the …
Let $O(2n+\ell)$ be the group of orthogonal matrices of size $\left(2n+\ell\right)\times \left(2n+\ell\right)$ equipped with the probability distribution given by normalized Haar measure. We study the probability \begin{equation*} p_{2n}^{\left(\ell\right)} = \mathbb{P}\left[M_{2n} \, \mbox{has no real eigenvalues}\right], \end{equation*} where $M_{2n}$ is the $2n\times 2n$ left top minor of a $(2n+\ell)\times(2n+\ell)$ orthogonal matrix. We prove that this probability is given in terms of a determinant identity minus a weighted Hankel matrix of size $n\times n$ that depends on the truncation …
We present a new perspective of assessing the rates of convergence to the Gaussian and Poisson distributions in the Erdös-Kac theorem for additive arithmetic functions $ψ$ of a random integer $J_n$ uniformly distributed over $\{1,...,n\}$. Our approach is probabilistic, working directly on spaces of random variables without any use of Fourier analytic methods, and our $ψ$ is more general than those considered in the literature. Our main results are (i) bounds on the Kolmogorov distance and Wasserstein distance between the distribution of the normalized $ψ(J_n)$ and the standard Gaussian distri…
We study large deviations principles for $ N $ random processes on the lattice $ \Z^d $ with finite time horizon $ [0,β] $ under a symmetrised measure where all initial and terminal points are uniformly given by a random permutation. That is, given a permutation $ σ$ of $ N $ elements and a vector $ (x_1,...,x_N) $ of $ N $ initial points we let the random processes terminate in the points $ (x_{σ(1)},...,x_{σ(N)}) $ and then sum over all possible permutations and initial points, weighted with an initial distribution. There is a two-level random mechanism and we prove two-level large deviation…
We present a null model for single- and multi-layered complex systems constructed using homogeneous and isotropic random Gaussian maps. By means of a Kac-Rice formalism, we show that the mean number of fixed points can be calculated as the expectation of the absolute value of the characteristic polynomial for a product of independent Gaussian (Ginibre) matrices. Furthermore, using techniques from Random Matrix Theory, we show that the high-dimensional limit of our system has a third-order phase transition between a phase with a single fixed point and a phase with exponentially many fixed point…
Let $ \{φ_i(z;α)\}_{i=0}^\infty $, corresponding to $ α\in(-1,1) $, be orthonormal Geronimus polynomials. We study asymptotic behavior of the expected number of real zeros, say $ \mathbb E_n(α) $, of random polynomials \[ P_n(z) := \sum_{i=0}^nη_iφ_i(z;α), \] where $ η_0,\dots,η_n $ are i.i.d. standard Gaussian random variables. When $ α=0 $, $ φ_i(z;0)=z^i $ and $ P_n(z)$ are called Kac polynomials. In this case it was shown by Wilkins that $ \mathbb E_n(0)$ admits an asymptotic expansion of the form \[ \mathbb E_n(0) \sim \frac2π\log(n+1) + \sum_{p=0}^\infty A_p(n+1)^{-p} \] (Kac himself obt…
The zeros of the random Laurent series $1/μ- \sum_{j=1}^\infty c_j/z^j$, where each $c_j$ is an independent standard complex Gaussian, is known to correspond to the scaled eigenvalues of a particular additive rank 1 perturbation of a standard complex Gaussian matrix. For the corresponding random Maclaurin series obtained by the replacement $z \mapsto 1/z$, we show that these same zeros correspond to the scaled eigenvalues of a particular multiplicative rank 1 perturbation of a random unitary matrix. Since the correlation functions of the latter are known, by taking an appropriate limit the cor…
We compute the persistence for the $2d$-diffusion equation with random initial condition, i.e., the probability $p_0(t)$ that the diffusion field, at a given point ${\bf x}$ in the plane, has not changed sign up to time $t$. For large $t$, we show that $p_0(t) \sim t^{-θ(2)}$ with $θ(2) = 3/16$. Using the connection between the $2d$-diffusion equation and Kac random polynomials, we show that the probability $q_0(n)$ that Kac polynomials, of (even) degree $n$, have no real root decays, for large $n$, as $q_0(n) \sim n^{-3/4}$. We obtain this result by using yet another connection with the trunc…
The book develops the fundamental ideas of the famous Kac-Rice formula for vectorvalued random fields. This formula allows to compute the expectation and moments of the measure, and integrals with respect to this measure, of the sets of levels of such fields. After a presentation of the historical context of the Kac-Rice formula, we give an elementary demonstration of the co-area formula. This formula replaces the change of variable formula in multiple integrals and a direct application of this formula gives the Kac-Rice formula for almost all levels. We emphasize the necessity of having the f…
