TY - JOUR
T1 - Topology trivialization transition in random non-gradient autonomous ODE's on a sphere
AU - Fyodorov, Yan V
PY - 2016/12/30
Y1 - 2016/12/30
N2 - We calculate the mean total number of equilibrium points in a system of N random autonomous ODE's introduced by Cugliandolo et al. in 1997 to describe non-relaxational glassy dynamics on the high-dimensional sphere. In doing it we suggest a new approach which allows such a calculation to be done most straightforwardly, and is based on efficiently incorporating the Langrange multiplier into the Kac-Rice framework. Analysing the asymptotic behaviour for large N we confirm that the phenomenon of 'topology trivialization' revealed earlier for other systems holds also in the present framework with nonrelaxational dynamics. Namely, by increasing the variance of the random 'magnetic field' term in dynamical equations we find a 'phase transition' from the exponentially abundant number of equilibria down to just two equilibria. Classifying the equilibria in the nontrivial phase by stability remains an open problem.
AB - We calculate the mean total number of equilibrium points in a system of N random autonomous ODE's introduced by Cugliandolo et al. in 1997 to describe non-relaxational glassy dynamics on the high-dimensional sphere. In doing it we suggest a new approach which allows such a calculation to be done most straightforwardly, and is based on efficiently incorporating the Langrange multiplier into the Kac-Rice framework. Analysing the asymptotic behaviour for large N we confirm that the phenomenon of 'topology trivialization' revealed earlier for other systems holds also in the present framework with nonrelaxational dynamics. Namely, by increasing the variance of the random 'magnetic field' term in dynamical equations we find a 'phase transition' from the exponentially abundant number of equilibria down to just two equilibria. Classifying the equilibria in the nontrivial phase by stability remains an open problem.
UR - http://iopscience.iop.org/article/10.1088/1742-5468/aa511a/meta
M3 - Article
VL - 2016
JO - Journal of Statistical Mechanics: Theory and Experiment
JF - Journal of Statistical Mechanics: Theory and Experiment
SN - 1742-5468
M1 - 124003
ER -
TY - JOUR
T1 - Towards rigorous analysis of the Levitov-Mirlin-Evers recursion
AU - Fyodorov, Yan V
AU - Kupiainen, Antti
AU - Webb, Christian
PY - 2016/11/10
Y1 - 2016/11/10
N2 - This paper aims to develop a rigorous asymptotic analysis of an approximate renormalization group recursion for inverse participation ratios Pq of critical powerlaw random band matrices. The recursion goes back to the work by Mirlin and Evers [37] and earlier works by Levitov [32, 33] and is aimed to describe the ensuing multifractality of the eigenvectors of such matrices. We point out both similarities and dissimilarities of LME recursion to those appearing in the theory of multiplicative cascades and branching random walks and show that the methods developed in those fields can be adapted to the present case. In particular the LME recursion is shown to exhibit a phase transition, which we expect is a freezing transition, where the role of temperature is played by the exponent q. However, the LME recursion has features that make its rigorous analysis considerably harder and we point out several open problems for further study
AB - This paper aims to develop a rigorous asymptotic analysis of an approximate renormalization group recursion for inverse participation ratios Pq of critical powerlaw random band matrices. The recursion goes back to the work by Mirlin and Evers [37] and earlier works by Levitov [32, 33] and is aimed to describe the ensuing multifractality of the eigenvectors of such matrices. We point out both similarities and dissimilarities of LME recursion to those appearing in the theory of multiplicative cascades and branching random walks and show that the methods developed in those fields can be adapted to the present case. In particular the LME recursion is shown to exhibit a phase transition, which we expect is a freezing transition, where the role of temperature is played by the exponent q. However, the LME recursion has features that make its rigorous analysis considerably harder and we point out several open problems for further study
U2 - 10.1088/0951-7715/29/12/3871
DO - 10.1088/0951-7715/29/12/3871
M3 - Article
VL - 29
JO - NONLINEARITY
JF - NONLINEARITY
SN - 0951-7715
IS - 12
M1 - 3871
ER -
TY - CHAP
T1 - Random Matrix Theory of resonances
T2 - An overview
AU - Fyodorov, Yan V.
PY - 2016/9/19
Y1 - 2016/9/19
N2 - Scattering of electromagnetic waves in billiard-like systems has become a standard experimental tool of studying properties associated with Quantum Chaos. Random Matrix Theory (RMT) describing statistics of eigenfrequencies and associated eigenfunctions remains one of the pillars of theoretical understanding of quantum chaotic systems. In a scattering system coupling to continuum via antennae converts real eigenfrequencies into poles of the scattering matrix in the complex frequency plane and the associated eigenfunctions into decaying resonance states. Understanding statistics of these poles, as well as associated non-orthogonal resonance eigenfunctions within RMT approach is still possible, though much more challenging task.
AB - Scattering of electromagnetic waves in billiard-like systems has become a standard experimental tool of studying properties associated with Quantum Chaos. Random Matrix Theory (RMT) describing statistics of eigenfrequencies and associated eigenfunctions remains one of the pillars of theoretical understanding of quantum chaotic systems. In a scattering system coupling to continuum via antennae converts real eigenfrequencies into poles of the scattering matrix in the complex frequency plane and the associated eigenfunctions into decaying resonance states. Understanding statistics of these poles, as well as associated non-orthogonal resonance eigenfunctions within RMT approach is still possible, though much more challenging task.
UR - http://www.scopus.com/inward/record.url?scp=84992128522&partnerID=8YFLogxK
U2 - 10.1109/URSI-EMTS.2016.7571486
DO - 10.1109/URSI-EMTS.2016.7571486
M3 - Other chapter contribution
AN - SCOPUS:84992128522
SP - 666
EP - 669
BT - 2016 URSI International Symposium on Electromagnetic Theory, EMTS 2016
PB - Institute of Electrical and Electronics Engineers Inc.
ER -
TY - JOUR
T1 - On the distribution of the maximum value of the characteristic polynomial of GUE random matrices
AU - Fyodorov, Yan V
AU - J Simm, N
PY - 2016/8/10
Y1 - 2016/8/10
N2 - Motivated by recently discovered relations between logarithmically correlated Gaussian processes and characteristic polynomials of large random $N\times N$ matrices H from the Gaussian unitary ensemble (GUE), we consider the problem of characterising the distribution of the global maximum of ${{D}_{N}}(x):=\log |\det (xI-H)|$ as $N\to \infty $ and $x\in (-1,1)$ . We arrive at an explicit expression for the asymptotic probability density of the (appropriately shifted) maximum by combining the rigorous Fisher–Hartwig asymptotics due to Krasovsky [34] with the heuristic freezing transition scenario for logarithmically correlated processes. Although the general idea behind the method is the same as for the earlier considered case of the circular unitary ensemble, the present GUE case poses new challenges. In particular we show how the conjectured self-duality in the freezing scenario plays the crucial role in our selection of the form of the maximum distribution. Finally, we demonstrate a good agreement of the found probability density with the results of direct numerical simulations of the maxima of D N (x).
AB - Motivated by recently discovered relations between logarithmically correlated Gaussian processes and characteristic polynomials of large random $N\times N$ matrices H from the Gaussian unitary ensemble (GUE), we consider the problem of characterising the distribution of the global maximum of ${{D}_{N}}(x):=\log |\det (xI-H)|$ as $N\to \infty $ and $x\in (-1,1)$ . We arrive at an explicit expression for the asymptotic probability density of the (appropriately shifted) maximum by combining the rigorous Fisher–Hartwig asymptotics due to Krasovsky [34] with the heuristic freezing transition scenario for logarithmically correlated processes. Although the general idea behind the method is the same as for the earlier considered case of the circular unitary ensemble, the present GUE case poses new challenges. In particular we show how the conjectured self-duality in the freezing scenario plays the crucial role in our selection of the form of the maximum distribution. Finally, we demonstrate a good agreement of the found probability density with the results of direct numerical simulations of the maxima of D N (x).
U2 - 10.1088/0951-7715/29/9/2837
DO - 10.1088/0951-7715/29/9/2837
M3 - Article
VL - 29
SP - 2837
EP - 2855
JO - NONLINEARITY
JF - NONLINEARITY
SN - 0951-7715
IS - 9
ER -
TY - JOUR
T1 - Nonlinear analogue of the May−Wigner instability transition
AU - Fyodorov, Yan V.
AU - Khoruzhenko, Boris A.
PY - 2016/6
Y1 - 2016/6
N2 - We study a system of N 1 degrees of freedom coupled via a smooth homogeneous Gaussian vector field with both gradient and divergence-free components. In the absence of coupling, the system is exponentially relaxing to an equilibrium with rate µ. We show that, while increasing the ratio of the coupling strength to the relaxation rate, the system experiences an abrupt transition from a topologically trivial phase portrait with a single equilibrium into a topologically non-trivial regime characterised by an exponential number of equilibria, the vast majority of which are expected to be unstable. It is suggested that this picture provides a global view on the nature of the May-Wigner instability transition originally discovered by local linear stability analysis.
AB - We study a system of N 1 degrees of freedom coupled via a smooth homogeneous Gaussian vector field with both gradient and divergence-free components. In the absence of coupling, the system is exponentially relaxing to an equilibrium with rate µ. We show that, while increasing the ratio of the coupling strength to the relaxation rate, the system experiences an abrupt transition from a topologically trivial phase portrait with a single equilibrium into a topologically non-trivial regime characterised by an exponential number of equilibria, the vast majority of which are expected to be unstable. It is suggested that this picture provides a global view on the nature of the May-Wigner instability transition originally discovered by local linear stability analysis.
U2 - 10.1073/pnas.1601136113
DO - 10.1073/pnas.1601136113
M3 - Article
VL - 113
SP - 6827
EP - 6832
JO - Proceedings of the National Academy of Sciences of the United States of America
JF - Proceedings of the National Academy of Sciences of the United States of America
SN - 0027-8424
IS - 25
ER -
TY - JOUR
T1 - Moments of the Position of the Maximum for GUE Characteristic Polynomials and for Log-Correlated Gaussian Processes
AU - Fyodorov, Yan V
AU - Le Doussal, Pierre
PY - 2016/5/19
Y1 - 2016/5/19
N2 - We study three instances of log-correlated processes on the interval: the logarithm of the Gaussian unitary ensemble (GUE) characteristic polynomial, the Gaussian log-correlated potential in presence of edge charges, and the Fractional Brownian motion with Hurst index H→0 (fBM0). In previous collaborations we obtained the probability distribution function (PDF) of the value of the global minimum (equivalently maximum) for the first two processes, using the freezing-duality conjecture (FDC). Here we study the PDF of the position of the maximum xm through its moments. Using replica, this requires calculating moments of the density of eigenvalues in the β-Jacobi ensemble. Using Jack polynomials we obtain an exact and explicit expression for both positive and negative integer moments for arbitrary β>0 and positive integer n in terms of sums over partitions. For positive moments, this expression agrees with a very recent independent derivation by Mezzadri and Reynolds. We check our results against a contour integral formula derived recently by Borodin and Gorin (presented in the Appendix 1 from these authors). The duality necessary for the FDC to work is proved, and on our expressions, found to correspond to exchange of partitions with their dual. Performing the limit n→0 and to negative Dyson index β→−2, we obtain the moments of xm and give explicit expressions for the lowest ones. Numerical checks for the GUE polynomials, performed independently by N. Simm, indicate encouraging agreement. Some results are also obtained for moments in Laguerre, Hermite-Gaussian, as well as circular and related ensembles. The correlations of the position and the value of the field at the minimum are also analyzed.
AB - We study three instances of log-correlated processes on the interval: the logarithm of the Gaussian unitary ensemble (GUE) characteristic polynomial, the Gaussian log-correlated potential in presence of edge charges, and the Fractional Brownian motion with Hurst index H→0 (fBM0). In previous collaborations we obtained the probability distribution function (PDF) of the value of the global minimum (equivalently maximum) for the first two processes, using the freezing-duality conjecture (FDC). Here we study the PDF of the position of the maximum xm through its moments. Using replica, this requires calculating moments of the density of eigenvalues in the β-Jacobi ensemble. Using Jack polynomials we obtain an exact and explicit expression for both positive and negative integer moments for arbitrary β>0 and positive integer n in terms of sums over partitions. For positive moments, this expression agrees with a very recent independent derivation by Mezzadri and Reynolds. We check our results against a contour integral formula derived recently by Borodin and Gorin (presented in the Appendix 1 from these authors). The duality necessary for the FDC to work is proved, and on our expressions, found to correspond to exchange of partitions with their dual. Performing the limit n→0 and to negative Dyson index β→−2, we obtain the moments of xm and give explicit expressions for the lowest ones. Numerical checks for the GUE polynomials, performed independently by N. Simm, indicate encouraging agreement. Some results are also obtained for moments in Laguerre, Hermite-Gaussian, as well as circular and related ensembles. The correlations of the position and the value of the field at the minimum are also analyzed.
U2 - 10.1007/s10955-016-1536-6
DO - 10.1007/s10955-016-1536-6
M3 - Article
VL - 164
SP - 190
EP - 240
JO - Journal of Statistical Physics
JF - Journal of Statistical Physics
SN - 0022-4715
ER -
TY - JOUR
T1 - Fractional Brownian motion with Hurst index H=0 and the Gaussian Unitary Ensemble
AU - Fyodorov, Y. V
AU - Khoruzhenko, B. A.
AU - Simm, N. J.
PY - 2016
Y1 - 2016
N2 - The goal of this paper is to establish a relation between characteristic polynomials of N×NN×N GUE random matrices HH as N→∞N→∞, and Gaussian processes with logarithmic correlations. We introduce a regularized version of fractional Brownian motion with zero Hurst index, which is a Gaussian process with stationary increments and logarithmic increment structure. Then we prove that this process appears as a limit of DN(z)=−log|det(H−zI)|DN(z)=−log|det(H−zI)| on mesoscopic scales as N→∞N→∞. By employing a Fourier integral representation, we use this to prove a continuous analogue of a result by Diaconis and Shahshahani [J. Appl. Probab. 31A (1994) 49–62]. On the macroscopic scale, DN(x)DN(x) gives rise to yet another type of Gaussian process with logarithmic correlations. We give an explicit construction of the latter in terms of a Chebyshev–Fourier random series.
AB - The goal of this paper is to establish a relation between characteristic polynomials of N×NN×N GUE random matrices HH as N→∞N→∞, and Gaussian processes with logarithmic correlations. We introduce a regularized version of fractional Brownian motion with zero Hurst index, which is a Gaussian process with stationary increments and logarithmic increment structure. Then we prove that this process appears as a limit of DN(z)=−log|det(H−zI)|DN(z)=−log|det(H−zI)| on mesoscopic scales as N→∞N→∞. By employing a Fourier integral representation, we use this to prove a continuous analogue of a result by Diaconis and Shahshahani [J. Appl. Probab. 31A (1994) 49–62]. On the macroscopic scale, DN(x)DN(x) gives rise to yet another type of Gaussian process with logarithmic correlations. We give an explicit construction of the latter in terms of a Chebyshev–Fourier random series.
U2 - 10.1214/15-AOP1039
DO - 10.1214/15-AOP1039
M3 - Article
VL - 44
SP - 2980
EP - 3031
JO - ANNALS OF PROBABILITY
JF - ANNALS OF PROBABILITY
SN - 0091-1798
IS - 4
ER -