Probability densities and distributions for spiked and general variance Wishart β\beta-ensembles

A Wishart matrix is said to be spiked when the underlying covariance matrix has a single eigenvalue $b$ different from unity. As $b$ increases through $b=2$, a gap forms from the largest eigenvalue to the rest of the spectrum, and with $b-2$ of order $N^{-1/3}$ the scaled largest eigenvalues form a well defined parameter dependent state. Recent works by Bloemendal and Vir\'ag [BV], and Mo, have quantified this parameter dependent state for real Wishart matrices from different viewpoints, and the former authors have done similarly for the spiked Wishart $\beta$-ensemble. The latter is defined in terms of certain random bidiagonal matrices. We use a recursive structure to give an alternative construction of the spiked and more generally the general variance Wishart $\beta$-ensemble, and we give the exact form of the joint eigenvalue PDF for the two matrices in the recurrence. In the case of real quaternion Wishart matrices ($\beta = 4$) the latter is recognised as having appeared in earlier studies on symmetrized last passage percolation, allowing the exact form of the scaled distribution of the largest eigenvalue to be given. This extends and simplifies earlier work of Wang, and is an alternative derivation to a result in [BV]. We also use the construction of the spiked Wishart $\beta$-ensemble from [BV] to give a simple derivation of the explicit form of the eigenvalue PDF.Comment: 18 page

Random walks and random fixed-point free involutions

A bijection is given between fixed point free involutions of $\{1,2,...,2N\}$ with maximum decreasing subsequence size $2p$ and two classes of vicious (non-intersecting) random walker configurations confined to the half line lattice points $l \ge 1$. In one class of walker configurations the maximum displacement of the right most walker is $p$. Because the scaled distribution of the maximum decreasing subsequence size is known to be in the soft edge GOE (random real symmetric matrices) universality class, the same holds true for the scaled distribution of the maximum displacement of the right most walker.Comment: 10 page

Recommendation Seeking Behavior: Empirical Study of Recommendation Needs in Everyday Life

This study explores why recommendation seekers look for recommendations, and how they interact with recommendations through their social milieu. This study utilizes qualitative one-week diary recordings and post-diary interviews to collect rich data that reflect recommendation seekers’ interaction and evaluation strategies in real life issues. The results show that respondents needed recommendations when they are new to situation, wish for changes from a routine behavior, seek trustworthy options or better solutions, and need inspiration. Degree of recommenders’ understanding participants’ situation is more significant than that of sharing interest and similarity with recommenders

Symmetrized models of last passage percolation and non-intersecting lattice paths

It has been shown that the last passage time in certain symmetrized models of directed percolation can be written in terms of averages over random matrices from the classical groups $U(l)$, $Sp(2l)$ and $O(l)$. We present a theory of such results based on non-intersecting lattice paths, and integration techniques familiar from the theory of random matrices. Detailed derivations of probabilities relating to two further symmetrizations are also given.Comment: 21 pages, 5 figure

Chaos synchronization in generalized Lorenz systems and an application to image encryption

Examples of synchronization, pervasive throughout the natural world, are often awe-inspiring because they tend to transcend our intuition. Synchronization in chaotic dynamical systems, of which the Lorenz system is a quintessential example, is even more surprising because the very defining features of chaos include sensitive dependence on initial conditions. It is worth pursuing, then, the question of whether high-dimensional extensions of such a system also exhibit synchronization. This study investigates synchronization in a set of high-dimensional generalizations of the Lorenz system obtained from the inclusion of additional Fourier modes. Numerical evidence supports that these systems exhibit self-synchronization. An example application of this phenomenon to image encryption is also provided. Numerical experiments also suggest that there is much more to synchronization in these generalized Lorenz systems than self-synchronization; while setting the dimension of the driver system higher than that of the receiver system does not result in perfect synchrony, the smaller the dimensional difference between the two, the more closely the receiver system tends to follow the driver, leading to self-synchronization when their dimensions are equal. © 2021 The Author

Dynamics of a tagged particle in the asymmetric exclusion process with the step initial condition

The one-dimensional totally asymmetric simple exclusion process (TASEP) is considered. We study the time evolution property of a tagged particle in TASEP with the step-type initial condition. Calculated is the multi-time joint distribution function of its position. Using the relation of the dynamics of TASEP to the Schur process, we show that the function is represented as the Fredholm determinant. We also study the scaling limit. The universality of the largest eigenvalue in the random matrix theory is realized in the limit. When the hopping rates of all particles are the same, it is found that the joint distribution function converges to that of the Airy process after the time at which the particle begins to move. On the other hand, when there are several particles with small hopping rate in front of a tagged particle, the limiting process changes at a certain time from the Airy process to the process of the largest eigenvalue in the Hermitian multi-matrix model with external sources.Comment: 48 pages, 8 figure

The largest eigenvalue of rank one deformation of large Wigner matrices

The purpose of this paper is to establish universality of the fluctuations of the largest eigenvalue of some non necessarily Gaussian complex Deformed Wigner Ensembles. The real model is also considered. Our approach is close to the one used by A. Soshnikov in the investigations of classical real or complex Wigner Ensembles. It is based on the computation of moments of traces of high powers of the random matrices under consideration

Exact solution for the stationary Kardar-Parisi-Zhang equation

We obtain the first exact solution for the stationary one-dimensional Kardar-Parisi-Zhang equation. A formula for the distribution of the height is given in terms of a Fredholm determinant, which is valid for any finite time $t$. The expression is explicit and compact enough so that it can be evaluated numerically. Furthermore, by extending the same scheme, we find an exact formula for the stationary two-point correlation function.Comment: 9 pages, 3 figure

Systematic comparison between the generalized Lorenz equations and DNS in the two-dimensional Rayleigh–Bénard convection

The classic Lorenz equations were originally derived from the two-dimensional Rayleigh–Bénard convection system considering an idealized case with the lowest order of harmonics. Although the low-order Lorenz equations have traditionally served as a minimal model for chaotic and intermittent atmospheric motions, even the dynamics of the two-dimensional Rayleigh–Bénard convection system is not fully represented by the Lorenz equations, and such differences have yet to be clearly identified in a systematic manner. In this paper, the convection problem is revisited through an investigation of various dynamical behaviors exhibited by a two-dimensional direct numerical simulation (DNS) and the generalized expansion of the Lorenz equations (GELE) derived by considering additional higher-order harmonics in the spectral expansions of periodic solutions. Notably, GELE allows us to understand how nonlinear interactions among high-order modes alter the dynamical features of the Lorenz equations including fixed points, chaotic attractors, and periodic solutions. It is verified that numerical solutions of the DNS can be recovered from the solutions of GELE when we consider the system with sufficiently high-order harmonics. At the lowest order, the classic Lorenz equations are recovered from GELE. Unlike in the Lorenz equations, we observe limit tori, which are the multi-dimensional analog of limit cycles, in the solutions of the DNS and GELE at high orders. Initial condition dependency in the DNS and Lorenz equations is also discussed. The Lorenz equations are a simplified nonlinear dynamical system derived from the two-dimensional Rayleigh–Bénard (RB) convection problem. They have been one of the best-known examples in chaos theory due to the peculiar bifurcation and chaos behaviors. They are often regarded as the minimal chaotic model for describing the convection system and, by extension, weather. Such an interpretation is sometimes challenged due to the simplifying restriction of considering only a few harmonics in the derivation. This study loosens this restriction by considering additional high-order harmonics and derives a system we call the generalized expansion of the Lorenz equations (GELE). GELE allows us to study how solutions transition from the classic Lorenz equations to high-order systems comparable to a two-dimensional direct numerical simulation (DNS). This study also proposes mathematical formulations for a direct comparison between the Lorenz equations, GELE, and two-dimensional DNS as the system’s order increases. This work advances our understanding of the convection system by bridging the gap between the classic model of Lorenz and a more realistic convection syste