Characters and fusion rules forW-algebras via quantized Drinfeld-Sokolov reduction

Using the cohomological approach toW-algebras, we calculate characters and fusion coefficients for their representations obtained from modular invariant representations of affine algebras by the quantized Drinfeld-Sokolov reduction.Supported in part by Junior Fellowship from Harvard Society of FellowsSupported in part by NSF grants DMS-8802489 and DMS-9103792Supported in part by RIMS-91 ProjectCommunicated by A. Jaffe     

Supersymmetric Vertex Algebras

We define and study the structure of SUSY Lie conformal and vertex algebras. This leads to effective rules for computations with superfields. Supported in part by NSF grants DMS-0201017 and DMS-0501395.     

Limits on stability of positive molecular ions

It is shown that a positive diatomic molecule consisting of N electrons and two nuclei with charges Z ₁ and Z ₂ is unstable with respect to breakup into two atomic subsystems if the nuclear charges are sufficiently large. Bounds on the critical charge are obtained in the limit as N .Research supported in part by NSF grants DMS-8709805 and DMS-8808112.     

A low temperature expansion for classicalN-vector models. II. Renormalization group equations

This paper continues the analysis of the low temperature expansions for classicalN-vector models started in [1]. A main part of it is a derivation of renormalization group equations and a construction of their solutions. To do this we have to introduce “a fluctuation integral” connected with a next renormalization transformation, and to make its preliminary analysis. The results of the paper are summarized in theorems stating that the renormalization transformation preserves the space of densitites, or actions described inductively in [1]. This work has been partially supported by the NSF Grant DMS-9102639.     

A Stochastic Model Associated with Enskog Equation and Its Kinetic Limit

 We examine a system of particles in which the particles travel deterministically in between stochastic collisions. The collisions are elastic and occur with probability ɛ ^d when two particles are at a distance σ. When the number of particles N goes to infinity and Nɛ ^d goes to a nonzero constant, we show that the particle density converges to a solution of the Enskog Equation. Received: 29 January 2002 / Accepted: 30 July 2002 Published online: 14 November 2002 RID="*" ID="*" Research supported in part by NSF Grant DMS-0072666     

Tail estimates for one-dimensional random walk in random environment

Suppose that the integers are assigned i.i.d. random variables { _x } (taking values in the unit interval), which serve as an environment. This environment defines a random walk {X _k } (called a RWRE) which, when atx, moves one step to the right with probability _x , and one step to the left with probability 1- _x . Solomon (1975) determined the almost-sure asymptotic speed (=rate of escape) of a RWRE. For certain environment distributions where the drifts 2 _x -1 can take both positive and negative values, we show that the chance of the RWRE deviating below this speed has a polynomial rate of decay, and determine the exponent in this power law; for environments which allow only positive and zero drifts, we show that these large-deviation probabilities decay like exp(–Cn ^1/3). This differs sharply from the rates derived by Greven and den-Hollander (1994) for large deviation probabilities conditioned on the environment. As a by product we also provide precise tail and moment estimates for the total population size in a Branching Process with Random Environment.Partially supported by NSF DMS-9209712 and DMS-9403553 grants, by a US-ISRAEL BSF grant and by the S. and N. Grand research fund.Research partially supported by NSF grant # DMS-9404391 and a Junior Faculty Fellowship from the Regents of the University of California.Partially supported by NSF grant # DMS-9302709, by a US-Israel BSF grant and by the fund for promotion of research at the Technion.     

A Poisson Structure on Compact Symmetric Spaces

We present some basic results on a natural Poisson structure on any compact symmetric space. The symplectic leaves of this structure are related to the orbits of the corresponding real semisimple group on the complex flag manifold.Acknowledgements We thank Sam Evens for many useful discussions. The first author was partially supported by NSF grant DMS-0072520. The second author was partially supported by NSF(USA) grants DMS-0105195 and DMS-0072551 and by the HHY Physical Sciences Fund at the University of Hong Kong.     

On the Largest Eigenvalue of a Random Subgraph of the Hypercube

Let G be a random subgraph of the n-cube where each edge appears randomly and independently with probability p. We prove that the largest eigenvalue of the adjacency matrix of G is almost surely where (G) is the maximum degree of G and the o(1) term tends to zero as max(^1/2(G),np) tends to infinity.Research was supported in part by the NSF grant DMS-0103948.Research was supported in part by NSF grants DMS-0106589, CCR-9987845 and by the State of New Jersey.     

Schramm–Loewner Equations Driven by Symmetric Stable Processes

We consider shape, size and regularity of the hulls K _t of the chordal Schramm–Loewner evolution driven by a symmetric α-stable process. We obtain derivative estimates, show that the domains are H?lder domains, prove that K _t has Hausdorff dimension 1, and show that the trace is right-continuous with left limits almost surely. Research supported in part by NSF Grant DMS-0600206. Research supported in part by NSF Grants DMS-0501726 and DMS-0244408.     

On algebraic equations satisfied by hypergeometric correlators in WZW models. I

It is proven that integral expressions for conformal correlators insl(2) WZW model found in [SV] satisfy certain natural algebraic equations. This implies that the above integrals really take their values in spaces of conformal blocks.The second author was supported in part by the NSF grant DMS-9202280. The third author was supported in part by the NSF grant DMS-9203939     

Decay of Solutions of the Wave Equation in the Kerr Geometry

We consider the Cauchy problem for the massless scalar wave equation in the Kerr geometry for smooth initial data compactly supported outside the event horizon. We prove that the solutions decay in time in L ^∞ _loc. The proof is based on a representation of the solution as an infinite sum over the angular momentum modes, each of which is an integral of the energy variable ω on the real line. This integral representation involves solutions of the radial and angular ODEs which arise in the separation of variables. Research supported in part by the Deutsche Forschungsgemeinschaft. Research supported by NSERC grant #RGPIN 105490-2004. Research supported in part by the NSF, Grant No. DMS-010-3998. Research supported in part by the NSF, Grant No. 33-585-7510-2-30. An erratum to this article is available at .     

Holonomy and Skyrme's Model

In this paper we consider two generalizations of the Skyrme model. One is a variational problem for maps from a compact 3-manifold to a compact Lie group. The other is a variational problem for flat connections. We describe the path components of the configuration spaces of smooth fields for each of the variational problems. We prove that the invariants separating the path components are well-defined for (not necessarily smooth) fields with finite Skyrme energy. We prove that for every possible value of these invariants there exists a minimizer of the Skyrme functional. Throughout the paper we emphasize the importance of holonomy in the Skyrme model. Some of the results may be useful in other contexts. In particular, we define the holonomy of a distributionally flat L²_loc connection; the local developing maps for such connections need not be continuous.The first author was partially supported by NSF grant DMS-0204651.The second author was partially supported by NSF grants DMS-9970638, and DMS-0200670     

Exact Solution of the Six-Vertex Model with Domain Wall Boundary Conditions. Disordered Phase

The six-vertex model, or the square ice model, with domain wall boundary conditions (DWBC) has been introduced and solved for finite N by Korepin and Izergin. The solution is based on the Yang–Baxter equations and it represents the free energy in terms of an N × N Hankel determinant. Paul Zinn–Justin observed that the Izergin– Korepin formula can be re-expressed in terms of the partition function of a random matrix model with a nonpolynomial interaction. We use this observation to obtain the large N asymptotics of the six-vertex model with DWBC in the disordered phase. The solution is based on the Riemann–Hilbert approach and the Deift–Zhou nonlinear steepest descent method. As was noticed by Kuperberg, the problem of enumeration of alternating sign matrices (the ASM problem) is a special case of the six-vertex model. We compare the obtained exact solution of the six-vertex model with known exact results for the 1, 2, and 3 enumerations of ASMs, and also with the exact solution on the so-called free fermion line. We prove the conjecture of Zinn–Justin that the partition function of the six-vertex model with DWBC has the asymptotics, as N → ∞, and we find the exact value of the exponent κ.The first author is supported in part by the National Science Foundation (NSF) Grant DMS-0354962.     

Local Semicircle Law and Complete Delocalization for Wigner Random Matrices

We consider N × N Hermitian random matrices with independent identical distributed entries. The matrix is normalized so that the average spacing between consecutive eigenvalues is of order 1/N. Under suitable assumptions on the distribution of the single matrix element, we prove that, away from the spectral edges, the density of eigenvalues concentrates around the Wigner semicircle law on energy scales . Up to the logarithmic factor, this is the smallest energy scale for which the semicircle law may be valid. We also prove that for all eigenvalues away from the spectral edges, the ℓ ^∞-norm of the corresponding eigenvectors is of order O(N ^−1/2), modulo logarithmic corrections. The upper bound O(N ^−1/2) implies that every eigenvector is completely delocalized, i.e., the maximum size of the components of the eigenvector is of the same order as their average size. In the Appendix, we include a lemma by J. Bourgain which removes one of our assumptions on the distribution of the matrix elements. Supported by Sofja-Kovalevskaya Award of the Humboldt Foundation. On leave from Cambridge University, UK. Partially supported by NSF grant DMS-0602038.     

Quantum affine algebras and deformations of the Virasoro and 237-1237-1237-1

Using the Wakimoto realization of quantum affine algebras we define new Poisson algebras, which areq-deformations of the classicalW. We also define their free field realizations, i.e. homomorphisms into some Heisenberg-Poisson algebras. The formulas for these homomorphisms coincide with formulas for spectra of transfer-matrices in the corresponding quantum integrable models derived by the Bethe-Ansatz method.Partially supported by NSF grants DMS-9205303 and DMS-9296120.     

The triangle law for Lyapunov exponents of large random matrices

For products,A(t)·A(t–1)...A(1), of i.i.d.N×N random matrices, with i.i.d. entries, a triangle law governs theN distribution of Lyapunov exponents, much like Wigner's quarter-circle law governs the singular values ofA(1). Our proof requires finite fourth moments and a bounded density; the result was previously derived only in the Gaussian case.Research supported in part by NSF Grants DMS-8902156 and DMS-9196086     

Duality of Orthogonal and Symplectic Matrix Integrals and Quaternionic Feynman Graphs

We present an asymptotic expansion for quaternionic self-adjoint matrix integrals. The Feynman diagrams appearing in the expansion are ordinary ribbon graphs and their non-orientable counterparts. We show that the 2N×2N Gaussian Orthogonal Ensemble (GOE) and N×N Gaussian Symplectic Ensemble (GSE) have exactly the same expansion term by term, except that the contributions from graphs on a non-orientable surface with odd Euler characteristic carry the opposite sign. As an application, we give a new topological proof of the known duality for correlations of characteristic polynomials, demonstrating that this duality is equivalent to Poincaré duality of graphs drawn on a compact surface. Another consequence of our graphical expansion formula is a simple and simultaneous (re)derivation of the Central Limit Theorem for GOE, GUE (Gaussian Unitary Ensemble) and GSE: The three cases have exactly the same graphical limiting formula except for an overall constant that represents the type of the ensemble.Research supported by NSF Grant DMS-9971371 and the University of California, Davis.Research supported by the University of California, Davis.     

Correlation Functions for <Emphasis Type="Italic">β</Emphasis>=1 Ensembles of Matrices of Odd Size

Using the method of Tracy and Widom we rederive the correlation functions for β=1 Hermitian and real asymmetric ensembles of N×N matrices with N odd. This research was supported in part by the National Science Foundation (DMS-0801243).     

Conjugacies for Tiling Dynamical Systems

We consider tiling dynamical systems and topological conjugacies between them. We prove that the criterion of being of finite type is invariant under topological conjugacy. For substitution tiling systems under rather general conditions, including the Penrose and pinwheel systems, we show that substitutions are invertible and that conjugacies are generalized sliding block codes.Research supported in part by NSF Vigre Grant DMS-0091946Research supported in part by NSF Grant DMS-0071643 and Texas ARP Grant 003658-158Acknowledgement The authors are grateful for the support of the Banff International Research Station, at which we formulated and proved Theorem 3.     

Decay Rates and Probability Estimates¶for Massive Dirac Particles¶in the Kerr–Newman Black Hole Geometry

The Cauchy problem is considered for the massive Dirac equation in the non-extreme Kerr–Newman geometry, for smooth initial data with compact support outside the event horizon and bounded angular momentum. We prove that the Dirac wave function decays in at least at the rate t ^−5/6. For generic initial data, this rate of decay is sharp. We derive a formula for the probability p that the Dirac particle escapes to infinity. For various conditions on the initial data, we show that p = 0, 1 or 0 < p < 1. The proofs are based on a refined analysis of the Dirac propagator constructed in [4]. Received: 20 August 2001 / Accepted: 22 January 2002 RID="*" ID="*"Present address: NWF I – Mathematik, Universit?t Regensburg, 93040 Regensburg, Germany.?E-mail: felix.finster@mathematik.uni-regensburg.de RID="**" ID="**"Research supported by NSERC grant # RGPIN 105490-1998. RID="***" ID="***"Research supported in part by the NSF, Grant No. DMS-0103998. RID="****" ID="****"Research supported in part by the NSF, Grant No. 33-585-7510-2-30.