On the Hopfield model at the critical temperature

We study the Hopfield model at temperature 1, when thenumber M(N) of patterns grows a bit slower than N. We reach a goodunderstanding of the model whenever M(N)≤N/(log N)¹¹. For example, we show that if M(N)→∞, for two typical configurations σ ¹, σ ², (∑ _{i ≤ N} σ¹ _i σ² _i )² is close to NM(N). Received: 15 December 1999 / Revised version: 8 December 2000 / Published online: 23 August 2001     

Anharmonic Oscillators in the Complex Plane, $\boldsymbol{\mathcal{PT}}$-symmetry,and Real Eigenvalues

For integers m ≥ 3 and 1 ≤ ℓ ≤ m − 1, we study the eigenvalue problems − u ^″(z) + [( − 1)^ℓ(iz) ^m  − P(iz)]u(z) = λu(z) with the boundary conditions that u(z) decays to zero as z tends to infinity along the rays argz=-\fracp2±\frac(l+1)pm+2\arg z=-\frac{\pi}{2}\pm \frac{(\ell+1)\pi}{m+2} in the complex plane, where P is a polynomial of degree at most m − 1. We provide asymptotic expansions of the eigenvalues λ _n . Then we show that if the eigenvalue problem is PT\mathcal{PT}-symmetric, then the eigenvalues are all real and positive with at most finitely many exceptions. Moreover, we show that when gcd(m,l)=1\gcd(m,\ell)=1, the eigenvalue problem has infinitely many real eigenvalues if and only if one of its translations or itself is PT\mathcal{PT}-symmetric. Also, we will prove some other interesting direct and inverse spectral results.     

Random walk on upper triangular matrices mixes rapidly

We present an upper bound O(n ² ) for the mixing time of a simple random walk on upper triangular matrices. We show that this bound is sharp up to a constant, and find tight bounds on the eigenvalue gap. We conclude by applying our results to indicate that the asymmetric exclusion process on a circle indeed mixes more rapidly than the corresponding symmetric process. Received: 25 January 1999 / Revised version: 17 September 1999 / Published online: 14 June 2000     

New Expressions for Repeated Upper Tail Integrals of the Normal Distribution

Expressions are given for repeated upper tail integrals of the univariate normal density (and so also for the general Hermite function) for both positive and negative arguments. The expressions involve moments of the form E(x + i N) ⁿ and E1 / (x ² + N ²) ⁿ , where N is a unit normal random variable. Laplace transforms are provided for the Hermite functions and the moments. The practical use of these expressions is illustrated.     

Asymptotics of random sums of negatively dependent random variables in the presence of dominatedly varying tails

Let X ₁ , X ₂ , . . . be a sequence of negatively dependent and identically distributed random variables, and let N be a counting random variable independent of X _i ’s. In this paper, we study the asymptotics for the tail probability of the random sum S_N = ?_{k = 1}^N X_k {S_N} = \sum\nolimits_{k = 1}^N {{X_k}} in the presence of heavy tails. We consider the following three cases: (i) P(N > x) = o(P(X ₁  > x)), and the distribution function (d.f.) of X ₁ is dominatedly varying; (ii) P(X ₁  > x) = o(P(N > x)), and the d.f. of N is dominatedly varying; (iii) the tails of X ₁ and N are asymptotically comparable and dominatedly varying.     

Exponential decay rate of survival probability in a disastrous random environment

Summary. We study the exponential decay rate of the survival probability up to time t>0 of a random walker moving in Zopf; ^d in a temporally and spatially fluctuating random environment. When the random walker has a speed parameter κ>0, we investigate the influence of κ on the exponential decay rate λ(d,κ). In particular we prove that for any fixed d≥1, λ(d,κ) behaves like as logκ as κ↘0. Received: 21 May 1996 / In revised form: 2 February 1997     

Three-term asymptotics for the boundary layers of semilinear elliptic eigenvalue problems

We consider the nonlinear eigenvalue problem −Δu=λ f(u) in Ω u=0 on ∂Ω, where Ω is a ball or an annulus in R^N (N ≥ 2) and λ > 0 is a parameter. It is known that if λ >> 1, then the corresponding positive solution u_λ develops boundary layers under some conditions on f. We establish the asymptotic formulas for the slope of the boundary layers of u_λ with the exact second term and the ‘optimal’ estimate of the third term.     

Normal Subgroups of Hecke Groups <Emphasis Type="Italic">H</Emphasis>(<Emphasis Type="Italic">λ</Emphasis>)

Let λ ≥ 2 and let H(λ) be the Hecke group associated to λ. Also let H(λ)\U be the Riemann surface associated to the Hecke group H(λ). In this article, we study the even subgroup H _e (λ) and the power subgroups H ^m (λ) of the Hecke groups H(λ). We also study some genus 0 normal subgroups of finite index of H(λ). Finally, we discuss free normal subgroups of H(λ).     

Large deviations and concentration properties for ∇ϕ interface models

We consider the massless field with zero boundary conditions outside D _N ≡D∩ (ℤ ^d /N) (N∈ℤ⁺), D a suitable subset of ℝ ^d , i.e. the continuous spin Gibbs measure ℙ _N on ℝ ^ℤd/N with Hamiltonian given by H(ϕ) = ∑ _{x,y:|x−y|=1} V(ϕ(x) −ϕ(y)) and ϕ(x) = 0 for x∈D _N ^C . The interaction V is taken to be strictly convex and with bounded second derivative. This is a standard effective model for a (d + 1)-dimensional interface: ϕ represents the height of the interface over the base D _N . Due to the choice of scaling of the base, we scale the height with the same factor by setting ξ _N = ϕ/N. We study various concentration and relaxation properties of the family of random surfaces {ξ _N } and of the induced family of gradient fields ∇ ^N ξ _N as the discretization step 1/N tends to zero (N→∞). In particular, we prove a large deviation principle for {ξ _N } and show that the corresponding rate function is given by ∫ _D σ(∇u(x))dx, where σ is the surface tension of the model. This is a multidimensional version of the sample path large deviation principle. We use this result to study the concentration properties of ℙ _N under the volume constraint, i.e. the constraint that (1/N ^d ) ∑ _x∈DN ξ _N (x) stays in a neighborhood of a fixed volume v > 0, and the hard–wall constraint, i.e. ξ _N (x) ≥ 0 for all x. This is therefore a model for a droplet of volume v lying above a hard wall. We prove that under these constraints the field {ξ _N of rescaled heights concentrates around the solution of a variational problem involving the surface tension, as it would be predicted by the phenomenological theory of phase boundaries. Our principal result, however, asserts local relaxation properties of the gradient field {∇ ^N ξ _N (·)} to the corresponding extremal Gibbs states. Thus, our approach has little in common with traditional large deviation techniques and is closer in spirit to hydrodynamic limit type of arguments. The proofs have both probabilistic and analytic aspects. Essential analytic tools are ? _p estimates for elliptic equations and the theory of Young measures. On the side of probability tools, a central role is played by the Helffer–Sj?strand [31] PDE representation for continuous spin systems which we rewrite in terms of random walk in random environment and by recent results of T. Funaki and H. Spohn [25] on the structure of gradient fields. Received: 3 March 1999 / Revised version: 9 August 1999 / Published online: 30 March 2000     

Almost sure asymptotics for the continuous parabolic Anderson model

We consider the parabolic Anderson problem ∂ _t u = κΔu + ξ(x)u on ℝ₊×ℝ ^d with initial condition u(0,x) = 1. Here κ > 0 is a diffusion constant and ξ is a random homogeneous potential. We concentrate on the two important cases of a Gaussian potential and a shot noise Poisson potential. Under some mild regularity assumptions, we derive the second-order term of the almost sure asymptotics of u(t, 0) as t→∞. Received: 26 July 1999 / Revised version: 6 April 2000 / Published online: 22 November 2000     

Transversal fluctuations for increasing subsequences on the plane

Consider a realization of a Poisson process in ℝ² with intensity 1 and take a maximal up/right path from the origin to (N, N) consisting of line segments between the points, where maximal means that it contains as many points as possible. The number of points in such a path has fluctuations of order N ^χ, where χ = 1/3, [BDJ]. Here we show that typical deviations of a maximal path from the diagonal x = y is of order N ^ξ with ξ = 2/3. This is consistent with the scaling identity χ = 2ξ− 1 which is believed to hold in many random growth models. Received: 16 April 1999 / Revised version: 5 July 1999 / Published online: 14 February 2000     

Asymptotic behavior of eigenvalues of two-parameter nonlinear Sturm-Liouville problems

The nonlinear two-parameter Sturm-Liouville problemu ^"+μg(u)=λf(u) is studied for μ, λ>0. By using Ljusternik-Schnirelman theory on the general level set developed by Zeidler, we shall show the existence of ann-th variational eigenvalue λ=λ_n(μ). Furthermore, for specialf andg, the asymptotic formula of λ₁(μ)) as μ→∞ is established.     

Limit theorems for a recursive maximum process with location-dependent periodic intensity-parameter

We investigate the recursive sequence Z _n : =  max {Z _n − 1,λ(Z _n − 1)X _n } where X _n is a sequence of iid random variables with exponential distributions and λ is a periodic positive bounded measurable function. We prove that the Césaro mean of the sequence λ(Z _n ) converges toward the essential minimum of λ. Subsequently we apply this result and obtain a limit theorem for the distributions of the sequence Z _n . The resulting limit is a Gumbel distribution.     

Increment sizes of the principal value of Brownian local time

Let W be a standard Brownian motion, and define Y(t)= ∫₀ ^t ds/W(s) as Cauchy's principal value related to local time. We determine: (a) the modulus of continuity of Y in the sense of P. Lévy; (b) the large increments of Y. Received: 1 April 1999 / Revised version: 27 September 1999 / Published online: 14 June 2000     

Hölder continuity for spatial and path processes via spectral analysis

For ν(dθ), a σ-finite Borel measure on R ^d , we consider L ²(ν(dθ))-valued stochastic processes Y(t) with te property that Y(t)=y(t,·) where y(t,θ)=∫ ^t ₀ e ^{−λ(θ)( t − s )} dm(s,θ) and m(t,θ) is a continuous martingale with quadratic variation [m](t)=∫ ^t ₀ g(s,θ)ds. We prove timewise H?lder continuity and maximal inequalities for Y and use these results to obtain Hilbert space regularity for a class of superrocesses as well as a class of stochastic evolutions of the form dX=AXdt+GdW with W a cylindrical Brownian motion. Maximal inequalities and H?lder continuity results are also provenfor the path process _t (τ)≗Y(τt∧t). Received: 25 June 1999 / Revised version: 28 August 2000 /?Published online: 9 March 2001     

Upper bounds for the first dirichlet eigenvalue of a tube around an algebraic complex curve of ℂ<Emphasis Type="Italic">P</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>(<Emphasis Type="Italic">λ</Emphasis>)

We give an upper bound for the first Dirichlet eigenvalue of a tube around a complex curve P of ℂP ⁿ (λ) which depends only on the radius of the tube and the degrees of the polynomials defining P. The bound is sharp on a totally geodesic ℂP ¹(λ) and gives a gap between the eigenvalue of a tube around ℂP ¹(λ) and around other complex curves.     

A stability property of symplectic packing

We prove that for any closed symplectic 4-manifold (M,Ω) with [Ω]∈H ²(M, Q) there exists a number N ₀ such that for every N≥N ₀, (M,Ω) admits full symplectic packing by N equal balls. We also indicate how to compute this N ₀. Our approach is based on Donaldson's symplectic submanifold theorem and on tools from the framework of Taubes theory of Gromov invariants. Oblatum 9-I-1998 & 1-VII-1998 / Published online: 14 January 1999     

Sturm Sequences and Random Eigenvalue Distributions

This paper proposes that the study of Sturm sequences is invaluable in the numerical computation and theoretical derivation of eigenvalue distributions of random matrix ensembles. We first explore the use of Sturm sequences to efficiently compute histograms of eigenvalues for symmetric tridiagonal matrices and apply these ideas to random matrix ensembles such as the β-Hermite ensemble. Using our techniques, we reduce the time to compute a histogram of the eigenvalues of such a matrix from O(n ²+m) to O(mn) time where n is the dimension of the matrix and m is the number of bins (with arbitrary bin centers and widths) desired in the histogram (m is usually much smaller than n). Second, we derive analytic formulas in terms of iterated multivariate integrals for the eigenvalue distribution and the largest eigenvalue distribution for arbitrary symmetric tridiagonal random matrix models. As an example of the utility of this approach, we give a derivation of both distributions for the β-Hermite random matrix ensemble (for general β). Third, we explore the relationship between the Sturm sequence of a random matrix and its shooting eigenvectors. We show using Sturm sequences that assuming the eigenvector contains no zeros, the number of sign changes in a shooting eigenvector of parameter λ is equal to the number of eigenvalues greater than λ. Finally, we use the techniques presented in the first section to experimentally demonstrate a O(log n) growth relationship between the variance of histogram bin values and the order of the β-Hermite matrix ensemble. This research was supported by NSF Grant DMS–0411962.     

Many partition relations below density

We force 2 ^λ to be large, and for many pairs in the interval (λ, 2 ^λ ) a strong version of the polarized partition relations holds. We apply this to problems in general topology. For example, consistently, every 2 ^λ is the successor of a singular and for every Hausdorff regular space X, hd(X) ≤ s(X)⁺³, hL(X) ≤ s(X)⁺³ and better when s(X) is regular, via a halfgraph partition relations. For the case s(X) = ℵ ₀ we get hd(X), hL(X) ≤ N ₂.     

Families of Spectral Sets for Bernoulli Convolutions

We study the harmonic analysis of Bernoulli measures μ _λ , a one-parameter family of compactly supported Borel probability measures on the real line. The parameter λ is a fixed number in the open interval (0,1). The measures μ _λ may be understood in any one of the following three equivalent ways: as infinite convolution measures of a two-point probability distribution; as the distribution of a random power series; or as an iterated function system (IFS) equilibrium measure determined by the two transformations λ(x±1). For a given λ, we consider the harmonic analysis in the sense of Fourier series in the Hilbert space L ²(μ _λ ). For L ²(μ _λ ) to have infinite families of orthogonal complex exponential functions e ^2πis(⋅), it is known that λ must be a rational number of the form \fracm2n\frac{m}{2n}, where m is odd. We show that L²(m_\frac12n)L^{2}(\mu_{\frac{1}{2n}}) has a variety of Fourier bases; i.e. orthonormal bases of exponential functions. For some other rational values of λ, we exhibit maximal Fourier families that are not orthonormal bases.