TY - JOUR
T1 - Statistics of Extremes in Eigenvalue-Counting Staircases
AU - Fyodorov, Yan V.
AU - Le Doussal, Pierre
PY - 2020/5/29
Y1 - 2020/5/29
N2 - We consider the number NθA(θ) of eigenvalues eiθj of a random unitary matrix, drawn from CUEβ(N), in the interval θj∈[θA,θ]. The deviations from its mean, NθA(θ)-E[NθA(θ)], form a random process as function of θ. We study the maximum of this process, by exploiting the mapping onto the statistical mechanics of log-correlated random landscapes. By using an extended Fisher-Hartwig conjecture supplemented with the freezing duality conjecture for log-correlated fields, we obtain the cumulants of the distribution of that maximum for any β>0. It exhibits combined features of standard counting statistics of fermions (free for β=2 and with Sutherland-type interaction for β≠2) in an interval and extremal statistics of the fractional Brownian motion with Hurst index H=0. The β=2 results are expected to apply to the statistics of zeroes of the Riemann Zeta function.
AB - We consider the number NθA(θ) of eigenvalues eiθj of a random unitary matrix, drawn from CUEβ(N), in the interval θj∈[θA,θ]. The deviations from its mean, NθA(θ)-E[NθA(θ)], form a random process as function of θ. We study the maximum of this process, by exploiting the mapping onto the statistical mechanics of log-correlated random landscapes. By using an extended Fisher-Hartwig conjecture supplemented with the freezing duality conjecture for log-correlated fields, we obtain the cumulants of the distribution of that maximum for any β>0. It exhibits combined features of standard counting statistics of fermions (free for β=2 and with Sutherland-type interaction for β≠2) in an interval and extremal statistics of the fractional Brownian motion with Hurst index H=0. The β=2 results are expected to apply to the statistics of zeroes of the Riemann Zeta function.
UR - http://www.scopus.com/inward/record.url?scp=85085987949&partnerID=8YFLogxK
U2 - 10.1103/PhysRevLett.124.210602
DO - 10.1103/PhysRevLett.124.210602
M3 - Article
AN - SCOPUS:85085987949
VL - 124
JO - Physical Review Letters
JF - Physical Review Letters
SN - 0031-9007
IS - 21
M1 - 210602
ER -
TY - JOUR
T1 - Manifolds Pinned by a High-Dimensional Random Landscape
T2 - Hessian at the Global Energy Minimum
AU - Fyodorov, Yan V.
AU - Le Doussal, Pierre
PY - 2020/3/19
Y1 - 2020/3/19
N2 - We consider an elastic manifold of internal dimension d and length L pinned in a N dimensional random potential and confined by an additional parabolic potential of curvature μ. We are interested in the mean spectral density ρ(λ) of the Hessian matrix K at the absolute minimum of the total energy. We use the replica approach to derive the system of equations for ρ(λ) for a fixed Ld in the N→ ∞ limit extending d= 0 results of our previous work (Fyodorov et al. in Ann Phys 397:1–64, 2018). A particular attention is devoted to analyzing the limit of extended lattice systems by letting L→ ∞. In all cases we show that for a confinement curvature μ exceeding a critical value μc, the so-called “Larkin mass”, the system is replica-symmetric and the Hessian spectrum is always gapped (from zero). The gap vanishes quadratically at μ→ μc. For μ< μc the replica symmetry breaking (RSB) occurs and the Hessian spectrum is either gapped or extends down to zero, depending on whether RSB is 1-step or full. In the 1-RSB case the gap vanishes in all d as (μc-μ)4 near the transition. In the full RSB case the gap is identically zero. A set of specific landscapes realize the so-called “marginal cases” in d= 1 , 2 which share both feature of the 1-step and the full RSB solution, and exhibit some scale invariance. We also obtain the average Green function associated to the Hessian and find that at the edge of the spectrum it decays exponentially in the distance within the internal space of the manifold with a length scale equal in all cases to the Larkin length introduced in the theory of pinning.
AB - We consider an elastic manifold of internal dimension d and length L pinned in a N dimensional random potential and confined by an additional parabolic potential of curvature μ. We are interested in the mean spectral density ρ(λ) of the Hessian matrix K at the absolute minimum of the total energy. We use the replica approach to derive the system of equations for ρ(λ) for a fixed Ld in the N→ ∞ limit extending d= 0 results of our previous work (Fyodorov et al. in Ann Phys 397:1–64, 2018). A particular attention is devoted to analyzing the limit of extended lattice systems by letting L→ ∞. In all cases we show that for a confinement curvature μ exceeding a critical value μc, the so-called “Larkin mass”, the system is replica-symmetric and the Hessian spectrum is always gapped (from zero). The gap vanishes quadratically at μ→ μc. For μ< μc the replica symmetry breaking (RSB) occurs and the Hessian spectrum is either gapped or extends down to zero, depending on whether RSB is 1-step or full. In the 1-RSB case the gap vanishes in all d as (μc-μ)4 near the transition. In the full RSB case the gap is identically zero. A set of specific landscapes realize the so-called “marginal cases” in d= 1 , 2 which share both feature of the 1-step and the full RSB solution, and exhibit some scale invariance. We also obtain the average Green function associated to the Hessian and find that at the edge of the spectrum it decays exponentially in the distance within the internal space of the manifold with a length scale equal in all cases to the Larkin length introduced in the theory of pinning.
KW - First keyword
KW - More
KW - Second keyword
UR - http://www.scopus.com/inward/record.url?scp=85082872548&partnerID=8YFLogxK
U2 - 10.1007/s10955-020-02522-2
DO - 10.1007/s10955-020-02522-2
M3 - Article
AN - SCOPUS:85082872548
VL - 179
SP - 176
EP - 215
JO - Journal of Statistical Physics
JF - Journal of Statistical Physics
SN - 0022-4715
IS - 1
ER -
TY - JOUR
T1 - Manifolds in a high-dimensional random landscape
T2 - Complexity of stationary points and depinning
AU - Fyodorov, Yan V.
AU - Le Doussal, Pierre
PY - 2020/2/18
Y1 - 2020/2/18
N2 - We obtain explicit expressions for the annealed complexities associated, respectively, with the total number of (i) stationary points and (ii) local minima of the energy landscape for an elastic manifold with internal dimension d<4 embedded in a random medium of dimension N≫1 and confined by a parabolic potential with the curvature parameter μ. These complexities are found to both vanish at the critical value μc identified as the Larkin mass. For μ<μc the system is in complex phase corresponding to the replica symmetry breaking in its T=0 thermodynamics. The complexities vanish, respectively, quadratically (stationary points) and cubically (minima) at μc-. For d≥1 they admit a finite "massless" limit μ=0 which is used to provide an upper bound for the depinning threshold under an applied force.
AB - We obtain explicit expressions for the annealed complexities associated, respectively, with the total number of (i) stationary points and (ii) local minima of the energy landscape for an elastic manifold with internal dimension d<4 embedded in a random medium of dimension N≫1 and confined by a parabolic potential with the curvature parameter μ. These complexities are found to both vanish at the critical value μc identified as the Larkin mass. For μ<μc the system is in complex phase corresponding to the replica symmetry breaking in its T=0 thermodynamics. The complexities vanish, respectively, quadratically (stationary points) and cubically (minima) at μc-. For d≥1 they admit a finite "massless" limit μ=0 which is used to provide an upper bound for the depinning threshold under an applied force.
UR - http://www.scopus.com/inward/record.url?scp=85080103276&partnerID=8YFLogxK
U2 - 10.1103/PhysRevE.101.020101
DO - 10.1103/PhysRevE.101.020101
M3 - Article
AN - SCOPUS:85080103276
VL - 101
JO - PHYSICAL REVIEW E
JF - PHYSICAL REVIEW E
SN - 1539-3755
IS - 2
M1 - 020101(R)
ER -
TY - JOUR
T1 - Statistics of off-diagonal entries of Wigner K-matrix for chaotic wave systems with absorption
AU - Belga Fedeli, Sirio
AU - Fyodorov, Yan
PY - 2020/2/7
Y1 - 2020/2/7
UR - https://arxiv.org/abs/1905.04157
M3 - Article
JO - Journal Of Physics A-Mathematical And Theoretical
JF - Journal Of Physics A-Mathematical And Theoretical
SN - 1751-8113
ER -
TY - JOUR
T1 - Optimal wave fields for micromanipulation in complex scattering environments
AU - Horodynski, Michael
AU - Kühmayer, Matthias
AU - Brandstötter, Andre
AU - Pichler, Kevin
AU - Fyodorov, Yan V.
AU - Kuhl, Ulrich
AU - Rotter, Stefan
PY - 2019/11/18
Y1 - 2019/11/18
N2 - The manipulation of small objects with light has become an indispensable tool in many areas of research, ranging from physics to biology and medicine1–7. Here, we demonstrate how to implement micromanipulation at the optimal level of efficiency for arbitrarily shaped targets and inside complex environments such as disordered media. Our approach is to design wavefronts in the far field8–15 with optimal properties in the near field of the target to apply the strongest possible force, pressure or torque as well as to achieve the most efficient focus inside the target. This non-iterative technique only relies on a simple eigenvalue problem established from the system’s scattering matrix and its dependence on small shifts in specific target parameters (access to the near field of the target is not required). To illustrate this concept, we perform a proof-of-principle experiment in the microwave regime, fully confirming our predictions.
AB - The manipulation of small objects with light has become an indispensable tool in many areas of research, ranging from physics to biology and medicine1–7. Here, we demonstrate how to implement micromanipulation at the optimal level of efficiency for arbitrarily shaped targets and inside complex environments such as disordered media. Our approach is to design wavefronts in the far field8–15 with optimal properties in the near field of the target to apply the strongest possible force, pressure or torque as well as to achieve the most efficient focus inside the target. This non-iterative technique only relies on a simple eigenvalue problem established from the system’s scattering matrix and its dependence on small shifts in specific target parameters (access to the near field of the target is not required). To illustrate this concept, we perform a proof-of-principle experiment in the microwave regime, fully confirming our predictions.
UR - http://www.scopus.com/inward/record.url?scp=85075344639&partnerID=8YFLogxK
U2 - 10.1038/s41566-019-0550-z
DO - 10.1038/s41566-019-0550-z
M3 - Letter
AN - SCOPUS:85075344639
JO - Nature Photonics
JF - Nature Photonics
SN - 1749-4885
ER -
TY - JOUR
T1 - Separability gap and large-deviation entanglement criterion
AU - Czartowski, Jakub
AU - Szymański, Konrad
AU - Gardas, Bartłomiej
AU - Fyodorov, Yan V.
AU - Zyczkowski, Karol
PY - 2019/10/28
Y1 - 2019/10/28
N2 - For a given Hamiltonian H on a multipartite quantum system, one is interested in finding the energy E0 of its ground state. In the separability approximation, arising as a natural consequence of measurement in a separable basis, one looks for the minimal expectation value λmin- of H among all product states. For several concrete model Hamiltonians, we investigate the difference λmin - -E0, called the separability gap, which vanishes if the ground state has a product structure. In the generic case of a random Hermitian matrix of the Gaussian orthogonal ensemble, we find explicit bounds for the size of the gap which depend on the number of subsystems and hold with probability one. This implies an effective entanglement criterion applicable for any multipartite quantum system: If an expectation value of a typical observable of a given state is sufficiently distant from the average value, the state is almost surely entangled.
AB - For a given Hamiltonian H on a multipartite quantum system, one is interested in finding the energy E0 of its ground state. In the separability approximation, arising as a natural consequence of measurement in a separable basis, one looks for the minimal expectation value λmin- of H among all product states. For several concrete model Hamiltonians, we investigate the difference λmin - -E0, called the separability gap, which vanishes if the ground state has a product structure. In the generic case of a random Hermitian matrix of the Gaussian orthogonal ensemble, we find explicit bounds for the size of the gap which depend on the number of subsystems and hold with probability one. This implies an effective entanglement criterion applicable for any multipartite quantum system: If an expectation value of a typical observable of a given state is sufficiently distant from the average value, the state is almost surely entangled.
UR - http://www.scopus.com/inward/record.url?scp=85074440305&partnerID=8YFLogxK
U2 - 10.1103/PhysRevA.100.042326
DO - 10.1103/PhysRevA.100.042326
M3 - Article
AN - SCOPUS:85074440305
VL - 100
JO - Physical Review A (Atomic, Molecular and Optical Physics)
JF - Physical Review A (Atomic, Molecular and Optical Physics)
SN - 1050-2947
IS - 4
M1 - 042326
ER -
TY - CHAP
T1 - Optimal light fields for micromanipulation in complex scattering environments
AU - Horodynski, M.
AU - Kühmayer, M.
AU - Brandstötter, A.
AU - Pichler, K.
AU - Fyodorov, Y. V.
AU - Kuhl, U.
AU - Rotter, S.
PY - 2019/9/8
Y1 - 2019/9/8
N2 - We demonstrate both theoretically and experimentally how to achieve wave states that are optimal for transferring momentum, torque, etc. on a target of arbitrary shape embedded in an arbitrary environment.
AB - We demonstrate both theoretically and experimentally how to achieve wave states that are optimal for transferring momentum, torque, etc. on a target of arbitrary shape embedded in an arbitrary environment.
UR - http://www.scopus.com/inward/record.url?scp=85086312678&partnerID=8YFLogxK
U2 - 10.1364/FIO.2019.FW6B.3
DO - 10.1364/FIO.2019.FW6B.3
M3 - Conference paper
T3 - Frontiers in Optics - Proceedings Frontiers in Optics + Laser Science APS/DLS
BT - Frontiers in Optics - Proceedings Frontiers in Optics + Laser Science APS/DLS
PB - Optical Society of America (OSA)
ER -
TY - JOUR
T1 - A Spin Glass Model for Reconstructing Nonlinearly Encrypted Signals Corrupted by Noise
AU - Fyodorov, Yan V
PY - 2019/6/15
Y1 - 2019/6/15
N2 - We deﬁne a (symmetric key) encryption of a signal s∈ RN as a random mapping s→y= (y1,...,yM)T ∈ RM known both to the sender and a recipient. In general the recipients may have access only to images y corrupted by an additive noise of unknown strength. Given the encryption redundancy parameter (ERP) μ = M/N ≥ 1 and the signal strength parameter R =i s2 i /N, we consider the problem of reconstructing s from its corrupted image by a Least Square Scheme for a certain class of random Gaussian mappings. The problem is equivalent to ﬁnding the conﬁguration of minimal energy in a certain version of spherical spin glass model, with squared Gaussian random interaction potential. We use the Parisi replica symmetry breaking scheme to evaluate the mean overlap p∞ ∈[ 0,1] between the original signal and its recovered image(known as’estimate’) as N →∞,for a given(’bare’)noise-to-signal ratio (NSR) γ ≥0. Such an overlap is a measure of the quality of the signal reconstruction. We explicitly analyze the general case of linear-quadratic family of random mappings and discuss the full p∞(γ) curve. When nonlinearity exceeds a certain threshold but redundancy is not yet too big, the replica symmetric solution is necessarily broken in someintervalofNSR.Weshowthatencryptionswithanonvanishinglinearcomponentpermit reconstructions with p∞ > 0 for anyμ>1 and anyγ<∞, with p∞ ∼ γ−1/2 as γ →∞ . In contrast, for the case of purely quadratic nonlinearity, for any ERP μ>1 there exists a threshold NSR value γc(μ) such that p∞ = 0 forγ>γ c(μ) making the reconstruction impossible. The behaviour close to the threshold is given
AB - We deﬁne a (symmetric key) encryption of a signal s∈ RN as a random mapping s→y= (y1,...,yM)T ∈ RM known both to the sender and a recipient. In general the recipients may have access only to images y corrupted by an additive noise of unknown strength. Given the encryption redundancy parameter (ERP) μ = M/N ≥ 1 and the signal strength parameter R =i s2 i /N, we consider the problem of reconstructing s from its corrupted image by a Least Square Scheme for a certain class of random Gaussian mappings. The problem is equivalent to ﬁnding the conﬁguration of minimal energy in a certain version of spherical spin glass model, with squared Gaussian random interaction potential. We use the Parisi replica symmetry breaking scheme to evaluate the mean overlap p∞ ∈[ 0,1] between the original signal and its recovered image(known as’estimate’) as N →∞,for a given(’bare’)noise-to-signal ratio (NSR) γ ≥0. Such an overlap is a measure of the quality of the signal reconstruction. We explicitly analyze the general case of linear-quadratic family of random mappings and discuss the full p∞(γ) curve. When nonlinearity exceeds a certain threshold but redundancy is not yet too big, the replica symmetric solution is necessarily broken in someintervalofNSR.Weshowthatencryptionswithanonvanishinglinearcomponentpermit reconstructions with p∞ > 0 for anyμ>1 and anyγ<∞, with p∞ ∼ γ−1/2 as γ →∞ . In contrast, for the case of purely quadratic nonlinearity, for any ERP μ>1 there exists a threshold NSR value γc(μ) such that p∞ = 0 forγ>γ c(μ) making the reconstruction impossible. The behaviour close to the threshold is given
KW - Inference
KW - Signal reconstruction
KW - Spin glass
UR - http://www.scopus.com/inward/record.url?scp=85060084854&partnerID=8YFLogxK
U2 - 10.1007/s10955-018-02217-9
DO - 10.1007/s10955-018-02217-9
M3 - Article
VL - 175
SP - 789
EP - 818
JO - Journal of Statistical Physics
JF - Journal of Statistical Physics
SN - 0022-4715
IS - 5
ER -
TY - JOUR
T1 - Reflection time difference as a probe of S-matrix zeroes in chaotic resonance scattering
AU - Fyodorov, Y. V.
PY - 2019/1/1
Y1 - 2019/1/1
N2 - Motivated by recent interest in the zeroes of S-matrix entries in the complex energy plane, the Heidelberg model of resonance scattering is used to introduce the notion of reflection time difference. As is shown, it plays the same role for the zeroes as the Wigner time delay plays for the S-matrix poles.
AB - Motivated by recent interest in the zeroes of S-matrix entries in the complex energy plane, the Heidelberg model of resonance scattering is used to introduce the notion of reflection time difference. As is shown, it plays the same role for the zeroes as the Wigner time delay plays for the S-matrix poles.
UR - http://www.scopus.com/inward/record.url?scp=85077469280&partnerID=8YFLogxK
U2 - 10.12693/APhysPolA.136.785
DO - 10.12693/APhysPolA.136.785
M3 - Article
AN - SCOPUS:85077469280
VL - 136
SP - 785
EP - 789
JO - ACTA PHYSICA POLONICA SERIES A
JF - ACTA PHYSICA POLONICA SERIES A
SN - 0587-4246
IS - 5
ER -
TY - JOUR
T1 - Hessian spectrum at the global minimum of high-dimensional random landscapes
AU - Fyodorov, Yan V
AU - Le Doussal, Pierre
PY - 2018/10/26
Y1 - 2018/10/26
N2 - Using the replica method we calculate the mean spectral density of the Hessian matrix at the global minimum of a random $N \gg 1$ dimensional isotropic, translationally invariant Gaussian random landscape confined by a parabolic potential with fixed curvature $\mu>0$. Simple landscapes with generically a single minimum are typical for $\mu>\mu_{c}$, and we show that the Hessian at the global minimum is always {\it gapped}, with the low spectral edge being strictly positive. When approaching from above the transitional point $\mu= \mu_{c}$ separating simple landscapes from ''glassy'' ones, with exponentially abundant minima, the spectral gap vanishes as $(\mu-\mu_c)^2$. For $\mu<\mu_c$
the Hessian spectrum is qualitatively different for 'moderately complex' and 'genuinely complex' landscapes. The former are typical for short-range correlated random potentials and correspond to 1-step replica-symmetry breaking mechanism. Their Hessian spectra turn out to be again gapped, with the gap vanishing on approaching $\mu_c$ from below with a larger critical exponent, as $(\mu_c-\mu)^4$. At the same time in the '' most complex'' landscapes with long-ranged power-law correlations the replica symmetry is completely broken. We show that in that case
the Hessian remains gapless for all values of $\mu<\mu_c$, indicating the presence of 'marginally stable' spatial
directions. Finally, the potentials with {\it logarithmic} correlations share both 1RSB nature and gapless spectrum.
The spectral density of the Hessian always takes the semi-circular form, up to a shift and an amplitude that we explicitly calculate.
AB - Using the replica method we calculate the mean spectral density of the Hessian matrix at the global minimum of a random $N \gg 1$ dimensional isotropic, translationally invariant Gaussian random landscape confined by a parabolic potential with fixed curvature $\mu>0$. Simple landscapes with generically a single minimum are typical for $\mu>\mu_{c}$, and we show that the Hessian at the global minimum is always {\it gapped}, with the low spectral edge being strictly positive. When approaching from above the transitional point $\mu= \mu_{c}$ separating simple landscapes from ''glassy'' ones, with exponentially abundant minima, the spectral gap vanishes as $(\mu-\mu_c)^2$. For $\mu<\mu_c$
the Hessian spectrum is qualitatively different for 'moderately complex' and 'genuinely complex' landscapes. The former are typical for short-range correlated random potentials and correspond to 1-step replica-symmetry breaking mechanism. Their Hessian spectra turn out to be again gapped, with the gap vanishing on approaching $\mu_c$ from below with a larger critical exponent, as $(\mu_c-\mu)^4$. At the same time in the '' most complex'' landscapes with long-ranged power-law correlations the replica symmetry is completely broken. We show that in that case
the Hessian remains gapless for all values of $\mu<\mu_c$, indicating the presence of 'marginally stable' spatial
directions. Finally, the potentials with {\it logarithmic} correlations share both 1RSB nature and gapless spectrum.
The spectral density of the Hessian always takes the semi-circular form, up to a shift and an amplitude that we explicitly calculate.
U2 - 10.1088/1751-8121/aae74f
DO - 10.1088/1751-8121/aae74f
M3 - Article
JO - Journal of Physics A
JF - Journal of Physics A
SN - 1751-8113
ER -
TY - JOUR
T1 - Extreme values of CUE characteristic polynomials
T2 - a numerical study
AU - Fyodorov, Yan V.
AU - Gnutzmann, Sven
AU - Keating, Jonathan P.
PY - 2018/10/22
Y1 - 2018/10/22
N2 - We present the results of systematic numerical computations relating to the extreme value statistics of the characteristic polynomials of random unitary matrices drawn from the circular unitary ensemble (CUE) of random matrix theory. In particular, we investigate a range of recent conjectures and theoretical results inspired by analogies with the theory of logarithmically-correlated Gaussian random fields. These include phenomena related to the conjectured freezing transition. Our numerical results are consistent with, and therefore support, the previous conjectures and theory. We also go beyond previous investigations in several directions: we provide the first quantitative evidence in support of a correlation between extreme values of the characteristic polynomials and large gaps in the spectrum, we investigate the rate of convergence to the limiting formulae previously considered, and we extend the previous analysis of the CUE to the CβE which corresponds to allowing the degree of the eigenvalue repulsion to become a parameter.
AB - We present the results of systematic numerical computations relating to the extreme value statistics of the characteristic polynomials of random unitary matrices drawn from the circular unitary ensemble (CUE) of random matrix theory. In particular, we investigate a range of recent conjectures and theoretical results inspired by analogies with the theory of logarithmically-correlated Gaussian random fields. These include phenomena related to the conjectured freezing transition. Our numerical results are consistent with, and therefore support, the previous conjectures and theory. We also go beyond previous investigations in several directions: we provide the first quantitative evidence in support of a correlation between extreme values of the characteristic polynomials and large gaps in the spectrum, we investigate the rate of convergence to the limiting formulae previously considered, and we extend the previous analysis of the CUE to the CβE which corresponds to allowing the degree of the eigenvalue repulsion to become a parameter.
KW - freezing transition
KW - log-correlated processes
KW - random matrix theory
UR - http://www.scopus.com/inward/record.url?scp=85055470867&partnerID=8YFLogxK
U2 - 10.1088/1751-8121/aae65a
DO - 10.1088/1751-8121/aae65a
M3 - Article
AN - SCOPUS:85055470867
VL - 51
JO - Journal of Physics A
JF - Journal of Physics A
SN - 1751-8113
IS - 46
M1 - 464001
ER -
TY - JOUR
T1 - On Statistics of Bi-Orthogonal Eigenvectors in Real and Complex Ginibre Ensembles: Combining Partial Schur Decomposition with Supersymmetry
AU - Fyodorov, Yan V
PY - 2018/6/11
Y1 - 2018/6/11
N2 - We suggest a method of studying the joint probability density (JPD) of an eigenvalue and the associated ‘non-orthogonality overlap factor’ (also known as the ‘eigenvalue condition number’) of the left and right eigenvectors for non-selfadjoint Gaussian random matrices of size N×N . First we derive the general finite N expression for the JPD of a real eigenvalue λ and the associated non-orthogonality factor in the real Ginibre ensemble, and then analyze its ‘bulk’ and ‘edge’ scaling limits. The ensuing distribution is maximally heavy-tailed, so that all integer moments beyond normalization are divergent. A similar calculation for a complex eigenvalue z and the associated non-orthogonality factor in the complex Ginibre ensemble is presented as well and yields a distribution with the finite first moment. Its ‘bulk’ scaling limit yields a distribution whose first moment reproduces the well-known result of Chalker and Mehlig (Phys Rev Lett 81(16):3367–3370, 1998), and we provide the ‘edge’ scaling distribution for this case as well. Our method involves evaluating the ensemble average of products and ratios of integer and half-integer powers of characteristic polynomials for Ginibre matrices, which we perform in the framework of a supersymmetry approach. Our paper complements recent studies by Bourgade and Dubach (The distribution of overlaps between eigenvectors of Ginibre matrices, 2018.
AB - We suggest a method of studying the joint probability density (JPD) of an eigenvalue and the associated ‘non-orthogonality overlap factor’ (also known as the ‘eigenvalue condition number’) of the left and right eigenvectors for non-selfadjoint Gaussian random matrices of size N×N . First we derive the general finite N expression for the JPD of a real eigenvalue λ and the associated non-orthogonality factor in the real Ginibre ensemble, and then analyze its ‘bulk’ and ‘edge’ scaling limits. The ensuing distribution is maximally heavy-tailed, so that all integer moments beyond normalization are divergent. A similar calculation for a complex eigenvalue z and the associated non-orthogonality factor in the complex Ginibre ensemble is presented as well and yields a distribution with the finite first moment. Its ‘bulk’ scaling limit yields a distribution whose first moment reproduces the well-known result of Chalker and Mehlig (Phys Rev Lett 81(16):3367–3370, 1998), and we provide the ‘edge’ scaling distribution for this case as well. Our method involves evaluating the ensemble average of products and ratios of integer and half-integer powers of characteristic polynomials for Ginibre matrices, which we perform in the framework of a supersymmetry approach. Our paper complements recent studies by Bourgade and Dubach (The distribution of overlaps between eigenvectors of Ginibre matrices, 2018.
U2 - 10.1007/s00220-018-3163-3
DO - 10.1007/s00220-018-3163-3
M3 - Article
JO - Communications in Mathematical Physics
JF - Communications in Mathematical Physics
SN - 0010-3616
ER -
TY - JOUR
T1 - On characteristic polynomials for a generalized chiral random matrix ensemble with a source
AU - Fyodorov, Yan V
AU - Grela, Jacek
AU - Strahov, Eugene
PY - 2018/3/5
Y1 - 2018/3/5
N2 - We evaluate averages involving characteristic polynomials, inverse characteristic polynomials and ratios of characteristic polynomials for a $N\times N$ random matrix taken from a L-deformed chiral Gaussian Unitary Ensemble with an external source Ω. Relation to a recently studied statistics of bi-orthogonal eigenvectors in the complex Ginibre ensemble, see Fyodorov (2017 arXiv:1710.04699), is briefly discussed as a motivation to study asymptotics of these objects in the case of external source proportional to the identity matrix. In particular, for an associated complex bulk/chiral edge scaling regime we retrieve the kernel related to Bessel/Macdonald functions.
AB - We evaluate averages involving characteristic polynomials, inverse characteristic polynomials and ratios of characteristic polynomials for a $N\times N$ random matrix taken from a L-deformed chiral Gaussian Unitary Ensemble with an external source Ω. Relation to a recently studied statistics of bi-orthogonal eigenvectors in the complex Ginibre ensemble, see Fyodorov (2017 arXiv:1710.04699), is briefly discussed as a motivation to study asymptotics of these objects in the case of external source proportional to the identity matrix. In particular, for an associated complex bulk/chiral edge scaling regime we retrieve the kernel related to Bessel/Macdonald functions.
U2 - 10.1088/1751-8121/aaae2a
DO - 10.1088/1751-8121/aaae2a
M3 - Article
VL - 51
JO - Journal Of Physics A-Mathematical And Theoretical
JF - Journal Of Physics A-Mathematical And Theoretical
SN - 1751-8113
IS - 13
ER -
TY - JOUR
T1 - Log-correlated random-energy models with extensive free-energy fluctuations:
T2 - Pathologies caused by rare events as signatures of phase transitions
AU - Cao, Xiangyu
AU - Fyodorov, Yan V
AU - Le Doussal, Pierre
PY - 2018/2/13
Y1 - 2018/2/13
N2 - We address systematically an apparent nonphysical behavior of the free-energy moment generating function for several instances of the logarithmically correlated models: the fractional Brownian motion with Hurst index H = 0 (fBm0) (and its bridge version), a one-dimensional model appearing in decaying Burgers turbulence with log-correlated initial conditions and, finally, the two-dimensional log-correlated random-energy model (logREM) introduced in Cao et al. [Phys. Rev. Lett. 118, 090601 (2017)] based on the two-dimensional Gaussian free field with background charges and directly related to the Liouville field theory. All these models share anomalously large fluctuations of the associated free energy, with a variance proportional to the log of the system size. We argue that a seemingly nonphysical vanishing of the moment generating function for some values of parameters is related to the termination point transition (i.e., prefreezing). We study the associated universal log corrections in the frozen phase, both for logREMs and for the standard REM, filling a gap in the literature. For the above mentioned integrable instances of logREMs, we predict the nontrivial free-energy cumulants describing non-Gaussian fluctuations on the top of the Gaussian with extensive variance. Some of the predictions are tested numerically.
AB - We address systematically an apparent nonphysical behavior of the free-energy moment generating function for several instances of the logarithmically correlated models: the fractional Brownian motion with Hurst index H = 0 (fBm0) (and its bridge version), a one-dimensional model appearing in decaying Burgers turbulence with log-correlated initial conditions and, finally, the two-dimensional log-correlated random-energy model (logREM) introduced in Cao et al. [Phys. Rev. Lett. 118, 090601 (2017)] based on the two-dimensional Gaussian free field with background charges and directly related to the Liouville field theory. All these models share anomalously large fluctuations of the associated free energy, with a variance proportional to the log of the system size. We argue that a seemingly nonphysical vanishing of the moment generating function for some values of parameters is related to the termination point transition (i.e., prefreezing). We study the associated universal log corrections in the frozen phase, both for logREMs and for the standard REM, filling a gap in the literature. For the above mentioned integrable instances of logREMs, we predict the nontrivial free-energy cumulants describing non-Gaussian fluctuations on the top of the Gaussian with extensive variance. Some of the predictions are tested numerically.
U2 - 10.1103/PhysRevE.97.022117
DO - 10.1103/PhysRevE.97.022117
M3 - Article
VL - 97
JO - PHYSICAL REVIEW E
JF - PHYSICAL REVIEW E
SN - 1539-3755
IS - 2
ER -
TY - BOOK
T1 - Stochastic processes and random matrices
T2 - Lecture notes of the Les Houches summer school
AU - Schehr, Grégory
AU - Altland, Alexander
AU - Fyodorov, Yan V.
AU - O'Connell, Neil
AU - Cugliandolo, Leticia F.
PY - 2018/1/18
Y1 - 2018/1/18
N2 - The field of stochastic processes and Random Matrix Theory (RMT) has been a rapidly evolving subject during the last fifteen years. The continuous development and discovery of new tools, connections and ideas have led to an avalanche of new results. These breakthroughs have been made possible thanks, to a large extent, to the recent development of various new techniques in RMT. Matrix models have been playing an important role in theoretical physics for a long time and they are currently also a very active domain of research in mathematics. An emblematic example of these recent advances concerns the theory of growth phenomena in the Kardar-Parisi-Zhang (KPZ) universality class where the joint efforts of physicists and mathematicians during the last twenty years have unveiled the beautiful connections between this fundamental problem of statistical mechanics and the theory of random matrices, namely the fluctuations of the largest eigenvalue of certain ensembles of random matrices. This text not only covers this topic in detail but also presents more recent developments that have emerged from these discoveries, for instance in the context of low dimensional heat transport (on the physics side) or integrable probability (on the mathematical side).
AB - The field of stochastic processes and Random Matrix Theory (RMT) has been a rapidly evolving subject during the last fifteen years. The continuous development and discovery of new tools, connections and ideas have led to an avalanche of new results. These breakthroughs have been made possible thanks, to a large extent, to the recent development of various new techniques in RMT. Matrix models have been playing an important role in theoretical physics for a long time and they are currently also a very active domain of research in mathematics. An emblematic example of these recent advances concerns the theory of growth phenomena in the Kardar-Parisi-Zhang (KPZ) universality class where the joint efforts of physicists and mathematicians during the last twenty years have unveiled the beautiful connections between this fundamental problem of statistical mechanics and the theory of random matrices, namely the fluctuations of the largest eigenvalue of certain ensembles of random matrices. This text not only covers this topic in detail but also presents more recent developments that have emerged from these discoveries, for instance in the context of low dimensional heat transport (on the physics side) or integrable probability (on the mathematical side).
UR - http://www.scopus.com/inward/record.url?scp=85052640972&partnerID=8YFLogxK
U2 - 10.1093/oso/9780198797319
DO - 10.1093/oso/9780198797319
M3 - Book
AN - SCOPUS:85052640972
SN - 9780198797319
VL - 104
BT - Stochastic processes and random matrices
PB - Oxford University Press
ER -
TY - JOUR
T1 - Exponential number of equilibria and depinning threshold for a directed polymer in a random potential
AU - Fyodorov, Yan V.
AU - Le Doussal, Pierre
AU - Rosso, Alberto
AU - Texier, Christophe
PY - 2018
Y1 - 2018
N2 - By extending the Kac-Rice approach to manifolds of finite internal dimension, we show that the mean number of all possible equilibria (i.e. force-free configurations, a.k.a. equilibrium points) of an elastic line (directed polymer), confined in a harmonic well and submitted to a quenched random Gaussian potential in dimension , grows exponentially with its length . The growth rate is found to be directly related to the generalised Lyapunov exponent (GLE) which is a moment-generating function characterising the large-deviation type fluctuations of the solution to the initial value problem associated with the random Schrödinger operator of the 1D Anderson localization problem. For strong confinement, the rate is small and given by a non-perturbative (instanton, Lifshitz tail-like) contribution to GLE. For weak confinement, the rate is found to be proportional to the inverse Larkin length of the pinning theory. As an application, identifying the depinning with a landscape “topology trivialization” phenomenon, we obtain an upper bound for the depinning threshold , in the presence of an applied force, for elastic lines and -dimensional manifolds, expressed through the mean modulus of the spectral determinant of the Laplace operators with a random potential. We also discuss the question of counting of stable equilibria. Finally, we extend the method to calculate the asymptotic number of equilibria at fixed energy (elastic, potential and total), and obtain the (annealed) distribution of the energy density over these equilibria (i.e. force-free configurations). Some connections with the Larkin model are also established.
AB - By extending the Kac-Rice approach to manifolds of finite internal dimension, we show that the mean number of all possible equilibria (i.e. force-free configurations, a.k.a. equilibrium points) of an elastic line (directed polymer), confined in a harmonic well and submitted to a quenched random Gaussian potential in dimension , grows exponentially with its length . The growth rate is found to be directly related to the generalised Lyapunov exponent (GLE) which is a moment-generating function characterising the large-deviation type fluctuations of the solution to the initial value problem associated with the random Schrödinger operator of the 1D Anderson localization problem. For strong confinement, the rate is small and given by a non-perturbative (instanton, Lifshitz tail-like) contribution to GLE. For weak confinement, the rate is found to be proportional to the inverse Larkin length of the pinning theory. As an application, identifying the depinning with a landscape “topology trivialization” phenomenon, we obtain an upper bound for the depinning threshold , in the presence of an applied force, for elastic lines and -dimensional manifolds, expressed through the mean modulus of the spectral determinant of the Laplace operators with a random potential. We also discuss the question of counting of stable equilibria. Finally, we extend the method to calculate the asymptotic number of equilibria at fixed energy (elastic, potential and total), and obtain the (annealed) distribution of the energy density over these equilibria (i.e. force-free configurations). Some connections with the Larkin model are also established.
KW - Directed polymer in random medium
KW - Pinning
KW - Random Schrödinger operator
KW - Anderson localisation
KW - Generalised Lyapunov exponent
U2 - 10.1016/j.aop.2018.07.029
DO - 10.1016/j.aop.2018.07.029
M3 - Article
VL - 397
SP - 1
EP - 64
JO - ANNALS OF PHYSICS
JF - ANNALS OF PHYSICS
SN - 0003-4916
ER -
TY - JOUR
T1 - Distribution of zeros of the S-matrix of chaotic cavities with localized losses and Coherent Perfect Absorption: non-perturbative results.
AU - Fyodorov, Yan V
AU - Suwunnarat, Suwun
AU - Kottos, Tsampikos
PY - 2017/6/13
Y1 - 2017/6/13
N2 - We employ the Random Matrix Theory framework to calculate the density of zeroes of an $M$-channel scattering matrix describing a chaotic cavity with a single localized absorber embedded in it. Our approach extends beyond the weak-coupling limit of the cavity with the channels and applies for any absorption strength. Importantly it provides an insight for the optimal amount of loss needed to realize a chaotic coherent perfect absorbing (CPA) trap. Our predictions are tested against simulations for two types of traps: a complex network of resonators and quantum graphs.
AB - We employ the Random Matrix Theory framework to calculate the density of zeroes of an $M$-channel scattering matrix describing a chaotic cavity with a single localized absorber embedded in it. Our approach extends beyond the weak-coupling limit of the cavity with the channels and applies for any absorption strength. Importantly it provides an insight for the optimal amount of loss needed to realize a chaotic coherent perfect absorbing (CPA) trap. Our predictions are tested against simulations for two types of traps: a complex network of resonators and quantum graphs.
U2 - 10.1088/1751-8121/aa793a
DO - 10.1088/1751-8121/aa793a
M3 - Article
VL - 50
JO - Journal Of Physics A-Mathematical And Theoretical
JF - Journal Of Physics A-Mathematical And Theoretical
SN - 1751-8113
IS - 30
ER -
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 -
TY - JOUR
T1 - Freezing transitions and extreme values:
T2 - random matrix theory, zeta (1/2+it) and disordered landscapes
AU - Fyodorov, Yan
AU - Keating, Jonathan P.
PY - 2014/1/28
Y1 - 2014/1/28
N2 - We argue that the freezing transition scenario, previously conjectured to occur in the statistical mechanics of 1/f-noise random energy models, governs, after reinterpretation, the value distribution of the maximum of the modulus of the characteristic polynomials pN(θ) of large N×N random unitary (circular unitary ensemble) matrices UN; i.e. the extreme value statistics of pN(θ) when N∞. In addition, we argue that it leads to multi-fractal-like behaviour in the total length μN(x) of the intervals in which |pN(θ)|>Nx,x>0, in the same limit. We speculate that our results extend to the large values taken by the Riemann zeta function ζ(s) over stretches of the critical line s=1/2+it of given constant length and present the results of numerical computations of the large values of s(1/2+it) . Our main purpose is to draw attention to the unexpected connections between these different extreme value problems.
AB - We argue that the freezing transition scenario, previously conjectured to occur in the statistical mechanics of 1/f-noise random energy models, governs, after reinterpretation, the value distribution of the maximum of the modulus of the characteristic polynomials pN(θ) of large N×N random unitary (circular unitary ensemble) matrices UN; i.e. the extreme value statistics of pN(θ) when N∞. In addition, we argue that it leads to multi-fractal-like behaviour in the total length μN(x) of the intervals in which |pN(θ)|>Nx,x>0, in the same limit. We speculate that our results extend to the large values taken by the Riemann zeta function ζ(s) over stretches of the critical line s=1/2+it of given constant length and present the results of numerical computations of the large values of s(1/2+it) . Our main purpose is to draw attention to the unexpected connections between these different extreme value problems.
U2 - 10.1098/rsta.2012.0503
DO - 10.1098/rsta.2012.0503
M3 - Article
JO - Philosophical transactions. Series A, Mathematical, physical, and engineering sciences
JF - Philosophical transactions. Series A, Mathematical, physical, and engineering sciences
SN - 1471-2962
ER -