The Experts below are selected from a list of 318 Experts worldwide ranked by ideXlab platform
Gernot Akemann - One of the best experts on this subject based on the ideXlab platform.
-
A singular-potential random matrix model arising in mean-field glassy systems
Physical Review E : Statistical Nonlinear and Soft Matter Physics, 2014Co-Authors: Gernot Akemann, Dario Villamaina, Pierpaolo VivoAbstract:We consider an invariant random matrix model where the standard Gaussian potential is distorted by an additional single pole of order $m$. We compute the average or macroscopic spectral density in the limit of large matrix size, solving the Loop Equation with the additional constraint of vanishing trace on average. The density is generally supported on two disconnected intervals lying on the two sides of the pole. In the limit of having no pole, we recover the standard semicircle. Obtained in the planar limit, our results apply to matrices with orthogonal, unitary or symplectic symmetry alike. The orthogonal case with $m=2$ is motivated by an application to spin glass physics. In the Sherrington-Kirkpatrick mean-field model, in the paramagnetic phase and for sufficiently large systems the spin glass susceptibility is a random variable, depending on the realization of disorder. It is essentially given by a linear statistics on the eigenvalues of the coupling matrix. As such its large deviation function can be computed using standard Coulomb fluid techniques. The resulting free energy of the associated fluid precisely corresponds to the partition function of our random matrix model. Numerical simulations provide an excellent confirmation of our analytical results.
-
macroscopic and microscopic non universalityof compact support random matrix theory
Nuclear Physics, 2000Co-Authors: Gernot Akemann, G VernizziAbstract:Abstract A random matrix model with a σ -model like constraint, the restricted trace ensemble (RTE), is solved in the large- n limit. In the macroscopic limit the smooth connected two-point resolvent G(z,w) is found to be non-universal, extending previous results from monomial to arbitrary polynomial potentials. Using Loop Equation techniques we give a closed though non-universal expression for G(z,w) , which extends recursively to all higher k -point resolvents. These findings are in contrast to the usual unconstrained one-matrix model. However, in the microscopic large- n limit, which probes only correlations at distance of the mean level spacing, we are able to show that the constraint does not modify the universal sine-law. In the case of monomial potentials V(M)=M 2p , we provide a relation valid for finite- n between the k -point correlation function of the RTE and the unconstrained model. In the microscopic large- n limit they coincide which proves the microscopic universality of RTEs.
-
higher genus correlators for the hermitian matrix model with multiple cuts
Nuclear Physics, 1996Co-Authors: Gernot AkemannAbstract:Abstract An iterative scheme is set up for solving the Loop Equation of the hermitian one-matrix model with a multi-cut structure. Explicit results are presented for genus one for an arbitrary but finite number of cuts. Due to the complicated form of the boundary conditions, the Loop correlators now contain elliptic integrals. This demonstrates the existence of new universality classes for the hermitian matrix model. The two-cut solution is investigated in more detail, including the double scaling limit. It is shown that in special cases it differs from the known continuum solution with one cut.
-
Higher genus correlators for the hermitian matrix model with multiple cuts
Nuclear Physics B, 1996Co-Authors: Gernot AkemannAbstract:An iterative scheme is set up for solving the Loop Equation of the hermitian one-matrix model with a multi-cut structure. Explicit results are presented for genus one for an arbitrary but finite number of cuts. Due to the complicated form of the boundary conditions, the Loop correlators now contain elliptic integrals. This demonstrates the existence of new universality classes for the hermitian matrix model. The two-cut solution is investigated in more detail, including the double-scaling limit. It is shown, that in special cases it differs from the known continuum solution with one cut.Comment: 25 pages, Latex file, 1 figure. One reference adde
-
Loop Equations for multi-cut matrix models
arXiv: High Energy Physics - Theory, 1995Co-Authors: Gernot AkemannAbstract:The Loop Equation for the complex one-matrix model with a multi-cut structure is derived and solved in the planar limit. An iterative scheme for higher genus contributions to the free energy and the multi-Loop correlators is presented for the two-cut model, where explicit results are given up to and including genus two. The double-scaling limit is analyzed and the relation to the one-cut solution of the hermitian and complex one-matrix model is discussed.
Marco Bochicchio - One of the best experts on this subject based on the ideXlab platform.
-
Quasi BPS Wilson Loops, localization of Loop Equation by homology and exact beta function in the large N limit of SU(N) Yang-Mills theory
Journal of High Energy Physics, 2009Co-Authors: Marco BochicchioAbstract:We localize the Loop Equation of large-N YM theory in the ASD variables on a critical Equation for an effective action by means of homological methods as opposed to the cohomological localization of equivariantly closed forms in local field theory. Our localization occurs for some special simple quasi BPS Wilson Loops, that have no perimeter divergence and no cusp anomaly for backtracking cusps, in a partial Eguchi-Kawai reduction from four to two dimensions of the non-commutative theory in the limit of infinite non-commutativity and in a lattice regularization in which the ASD integration variables live at the points of the lattice, thus implying an embedding of parabolic Higgs bundles in the YM functional integral. We find that the beta function of the effective action is saturated by the non-commutative ASD vortices of the EK reduction. An exact canonical beta function of NSVZ type that reproduces the universal first and second perturbative coefficients follows by the localization on vortices. Finally we argue that a scheme can be found in which the canonical coupling coincides with the physical charge between static quark sources in the large-N limit and we compare our theoretical calculation with some numerical lattice result.
-
quasi bps wilson Loops localization of Loop Equation by homology and exact beta function in the large n limit of su n yang mills theory
Journal of High Energy Physics, 2009Co-Authors: Marco BochicchioAbstract:We localize the Loop Equation of large-N YM theory in the anti-self-dual variables on a critical Equation for an effective action by means of homological methods as opposed to the cohomological localization of equivariantly closed forms in local field theory. Our localization occurs for some special simple quasi BPS Wilson Loops, that have no perimeter divergence and no cusp anomaly for backtracking cusps, in a partial Eguchi-Kawai reduction from four to two dimensions of the non-commutative theory in the limit of infinite non-commutativity and in a lattice regularization in which the anti-self-dual integration variables live at the points of the lattice, thus implying an embedding of parabolic Higgs bundles in the YM functional integral. We find that the beta function of the effective action is saturated by the non-commutative anti-self-dual vortices of the Eguchi-Kawai reduction. An exact canonical beta function of Novikov-Shifman-Vainshtein-Zakharov type, that reproduces the universal first and second perturbative coefficients follows by the localization on vortices. Finally we argue that a scheme can be found in which the canonical coupling coincides with the physical charge between static quark sources in the large-N limit and we compare our theoretical calculation with some numerical lattice result.
-
Exact beta function and glueball spectrum in large N Yang-Mills theory.
2009Co-Authors: Marco Bochicchio, Piazzale Aldo MoroAbstract:In the pure large-N Yang-Mills theory there is a quasi-BPS sector that is exactly solvable at large N. It follows an exact beta function and the glueball spectrum in this sector. The main technical tool is a new holomorphic Loop Equation for quasi-BPS Wilson Loops, that occurs as a nonsupersymmetric analogue of Dijkgraaf-Vafa holomorphic Loop Equation for the glueball superpotential of N = 1 SUSY gauge theories. The new holomorphic Loop Equation is localized, i.e. reduced to a critical Equation, by a deformation of the Loop that is a vanishing boundary in homology, somehow in analogy with Witten’s cohomological localization by a coboundary deformation in SUSY gauge theories.
-
Exact beta function from the holographic Loop Equation of large-N QCD4
Journal of High Energy Physics, 2007Co-Authors: Marco BochicchioAbstract:We construct and study a quantum holographic effective action, ?q, whose critical Equation implies the holographic Loop Equation of large-N QCD4 for planar self-avoiding Loops in a certain regularization scheme. We extract from ?q the exact beta function in the given scheme. For the Wilsonean coupling constant the beta function is exactly one Loop and the first coefficient, ?0, agrees with its value in perturbation theory. For the canonical coupling constant the exact beta function has a NSV Z form and the first two coefficients in powers of the coupling, ?0 and ?1, agree with their value in perturbation theory.
-
Twisted local systems solve the (holographic) Loop Equation of large-N QCD_4
Journal of High Energy Physics, 2005Co-Authors: Marco BochicchioAbstract:We construct a holographic map from the Loop Equation of large-N QCD in d=2 and d=4, for planar self-avoiding Loops, to the critical Equation of an equivalent effective action. The holographic map is based on two ingredients: an already proposed holographic form of the Loop Equation, such that the quantum contribution is reduced to a regularized residue; a new conformal map from the region encircled by the based Loop to a cuspidal fundamental domain in the upper half-plane, such that the regularized residue vanishes at the cusp. As a check, we study the first coefficient of the beta function and that part of the second coefficient which arises from the rescaling anomaly, in passing from the Wilsonian to the canonically normalised (holographic) effective action.
Shota Komatsu - One of the best experts on this subject based on the ideXlab platform.
-
Loop Equation and Exact Soft Anomalous Dimension in N=4 Super Yang-Mills
Journal of High Energy Physics, 2020Co-Authors: Simone Giombi, Shota KomatsuAbstract:BPS Wilson Loops in supersymmetric gauge theories have been the subjects of active research since they are often amenable to exact computation. So far most of the studies have focused on Loops that do not intersect. In this paper, we derive exact results for intersecting 1/8 BPS Wilson Loops in N=4 supersymmetric Yang-Mills theory, using a combination of supersymmetric localization and the Loop Equation in 2d gauge theory. The result is given by a novel matrix-model-like representation which couples multiple contour integrals and a Gaussian matrix model. We evaluate the integral at large N, and make contact with the string worldsheet description at strong coupling. As an application of our results, we compute exactly a small-angle limit (and more generally near-BPS limits) of the cross anomalous dimension which governs the UV divergence of intersecting Wilson lines. The same quantity describes the soft anomalous dimension of scattering amplitudes of W-bosons in the Coulomb branch.
-
Loop Equation and exact soft anomalous dimension in n mathcal n 4 super yang mills
Journal of High Energy Physics, 2020Co-Authors: Simone Giombi, Shota KomatsuAbstract:BPS Wilson Loops in supersymmetric gauge theories have been the subjects of active research since they are often amenable to exact computation. So far most of the studies have focused on Loops that do not intersect. In this paper, we derive exact results for intersecting 1/8 BPS Wilson Loops in $$ \mathcal{N} $$ = 4 supersymmetric Yang-Mills theory, using a combination of supersymmetric localization and the Loop Equation in 2d gauge theory. The result is given by a novel matrix-model-like representation which couples multiple contour integrals and a Gaussian matrix model. We evaluate the integral at large N , and make contact with the string worldsheet description at strong coupling. As an application of our results, we compute exactly a small-angle limit (and more generally near-BPS limits) of the cross anomalous dimension which governs the UV divergence of intersecting Wilson lines. The same quantity describes the soft anomalous dimension of scattering amplitudes of W -bosons in the Coulomb branch.
Chigak Itoi - One of the best experts on this subject based on the ideXlab platform.
-
universal wide correlators in non gaussian orthogonal unitary and symplectic random matrix ensembles
Nuclear Physics, 1997Co-Authors: Chigak ItoiAbstract:Abstract We calculate wide distance connected correlators in non-gaussian orthogonal, unitary and symplectic random matrix ensembles by solving the Loop Equation in the 1/ N expansion. The multilevel correlator is shown to be universal in the large- N limit. We show the algorithm to obtain the connected correlator to an arbitrary order in the 1/ N expansion.
-
universal wide correlators in non gaussian orthogonal unitary and symplectic random matrix ensembles
arXiv: Condensed Matter, 1996Co-Authors: Chigak ItoiAbstract:We calculate wide distance connected correlators in non-gaussian orthogonal, unitary and symplectic random matrix ensembles by solving the Loop Equation in the 1/N-expansion. The multi-level correlator is shown to be universal in large N limit. We show the algorithm to obtain the connected correlator to an arbitrary order in the 1/N-expansion.
Peter J. Forrester - One of the best experts on this subject based on the ideXlab platform.
-
Loop Equation Analysis of the Circular $ \beta $ Ensembles
Journal of High Energy Physics, 2015Co-Authors: N. S. Witte, Peter J. ForresterAbstract:We construct a hierarchy of Loop Equations for invariant circular ensembles. These are valid for general classes of potentials and for arbitrary inverse temperatures $ {\rm Re}\,\beta>0 $ and number of eigenvalues $ N $. Using matching arguments for the resolvent functions of linear statistics $ f(\zeta)=(\zeta+z)/(\zeta-z) $ in a particular asymptotic regime, the global regime, we systematically develop the corresponding large $ N $ expansion and apply this solution scheme to the Dyson circular ensemble. Currently we can compute the second resolvent function to ten orders in this expansion and also its general Fourier coefficient or moment $ m_{k} $ to an equivalent length. The leading large $ N $, large $ k $, $ k/N $ fixed form of the moments can be related to the small wave-number expansion of the structure function in the bulk, scaled Dyson circular ensemble, known from earlier work. From the moment expansion we conjecture some exact partial fraction forms for the low $ k $ moments. For all of the forgoing results we have made a comparison with the exactly soluble cases of $ \beta = 1,2,4 $, general $ N $ and even, positive $ \beta $, $ N=2,3 $.
-
Loop Equation analysis of the circular β ensembles
Journal of High Energy Physics, 2015Co-Authors: N. S. Witte, Peter J. ForresterAbstract:We construct a hierarchy of Loop Equations for invariant circular ensembles. These are valid for general classes of potentials and for arbitrary inverse temperatures Re β > 0 and number of eigenvalues N. Using matching arguments for the resolvent functions of linear statistics f(ζ) = (ζ + z)/(ζ − z) in a particular asymptotic regime, the global regime, we systematically develop the corresponding large N expansion and apply this solution scheme to the Dyson circular ensemble. Currently we can compute the second resolvent function to ten orders in this expansion and also its general Fourier coefficient or moment mk to an equivalent length. The leading large N, large k, k/N fixed form of the moments can be related to the small wave-number expansion of the structure function in the bulk, scaled Dyson circular ensemble, known from earlier work. From the moment expansion we conjecture some exact partial fraction forms for the low k moments. For all of the forgoing results we have made a comparison with the exactly soluble cases of β = 1, 2, 4, general N and even, positive β, N = 2, 3.
-
Loop Equation analysis of the circular beta ensembles
arXiv: Mathematical Physics, 2014Co-Authors: N. S. Witte, Peter J. ForresterAbstract:We construct a hierarchy of Loop Equations for invariant circular ensembles. These are valid for general classes of potentials and for arbitrary inverse temperatures $ {\rm Re}\,\beta>0 $ and number of eigenvalues $ N $. Using matching arguments for the resolvent functions of linear statistics $ f(\zeta)=(\zeta+z)/(\zeta-z) $ in a particular asymptotic regime, the global regime, we systematically develop the corresponding large $ N $ expansion and apply this solution scheme to the Dyson circular ensemble. Currently we can compute the second resolvent function to ten orders in this expansion and also its general Fourier coefficient or moment $ m_{k} $ to an equivalent length. The leading large $ N $, large $ k $, $ k/N $ fixed form of the moments can be related to the small wave-number expansion of the structure function in the bulk, scaled Dyson circular ensemble, known from earlier work. From the moment expansion we conjecture some exact partial fraction forms for the low $ k $ moments. For all of the forgoing results we have made a comparison with the exactly soluble cases of $ \beta = 1,2,4 $, general $ N $ and even, positive $ \beta $, $ N=2,3 $.
