Non-Intersecting Squared Bessel Paths and Multiple Orthogonal Polynomials for Modified Bessel Weights

We study a model of n non-intersecting squared Bessel processes in the confluent case: all paths start at time t = 0 at the same positive value x = a, remain positive, and are conditioned to end at time t = T at x = 0. In the limit n → ∞, after appropriate rescaling, the paths fill out a region in the tx-plane that we describe explicitly. In particular, the paths initially stay away from the hard edge at x = 0, but at a certain critical time t* the smallest paths hit the hard edge and from then on are stuck to it. For t ≠ t* we obtain the usual scaling limits from random matrix theory, namely the sine, Airy, and Bessel kernels. A key fact is that the positions of the paths at any time t constitute a multiple orthogonal polynomial ensemble, corresponding to a system of two modified Bessel-type weights. As a consequence, there is a 3 × 3 matrix valued Riemann-Hilbert problem characterizing this model, that we analyze in the large n limit using the Deift-Zhou steepest descent method. There are some novel ingredients in the Riemann-Hilbert analysis that are of independent interest.     

Non-Intersecting Squared Bessel Paths at a Hard-Edge Tacnode

The squared Bessel process is a 1-dimensional diffusion process related to the squared norm of a higher dimensional Brownian motion. We study a model of n non-intersecting squared Bessel paths, with all paths starting at the same point a > 0 at time t = 0 and ending at the same point b > 0 at time t = 1. Our interest lies in the critical regime ab = 1/4, for which the paths are tangent to the hard edge at the origin at a critical time ${t^*\in (0,1)}$ . The critical behavior of the paths for n → ∞ is studied in a scaling limit with time t = t ^* + O(n ^?1/3) and temperature T = 1 + O(n ^?2/3). This leads to a critical correlation kernel that is defined via a new Riemann-Hilbert problem of size 4 × 4. The Riemann-Hilbert problem gives rise to a new Lax pair representation for the Hastings-McLeod solution to the inhomogeneous Painlevé II equation q′′(x) = xq(x) + 2q ³(x) ? ν, where ν = α + 1/2 with α > ?1 the parameter of the squared Bessel process. These results extend our recent work with Kuijlaars and Zhang (Comm Pure Appl Math 64:1305–1383, 2011) for the homogeneous case ν = 0.     

Universality of a Double Scaling Limit near Singular Edge Points in Random Matrix Models

We consider unitary random matrix ensembles on the space of Hermitian n × n matrices M, where the confining potential V _s,t is such that the limiting mean density of eigenvalues (as n→∞ and s,t→ 0) vanishes like a power 5/2 at a (singular) endpoint of its support. The main purpose of this paper is to prove universality of the eigenvalue correlation kernel in a double scaling limit. The limiting kernel is built out of functions associated with a special solution of the P _I ² equation, which is a fourth order analogue of the Painlevé I equation. In order to prove our result, we use the well-known connection between the eigenvalue correlation kernel and the Riemann-Hilbert (RH) problem for orthogonal polynomials, together with the Deift/Zhou steepest descent method to analyze the RH problem asymptotically. The key step in the asymptotic analysis will be the construction of a parametrix near the singular endpoint, for which we use the model RH problem for the special solution of the P _I ² equation. In addition, the RH method allows us to determine the asymptotics (in a double scaling limit) of the recurrence coefficients of the orthogonal polynomials with respect to the varying weights on . The special solution of the P _I ² equation pops up in the n ^−2/7-term of the asymptotics.     

The corrections to scaling within Mazenko’s theory in the limit of low and high dimensions

We consider corrections to scaling within an approximate theory developed by Mazenko for nonconserved order parameter in the limit of low (d → 1) and high (d → ∞) dimensions. The corrections to scaling considered here follows from the departures of the initial condition from the scaling morphology. Including corrections to scaling, the equal time correlation function has the form: C(r, t) = f ₀(r/L)+L ^−ω f ₁(r/L)+…, where L is a characteristic length scale (i.e. domain size). The correction-to-scaling exponent ω and the correction-to-scaling functions f ₁(x) are calculated for both low and high dimensions. In both dimensions the value of ω is found to be ω = 4 similar to 1D Glauber model and OJK theory (the theory developed by Ohta, Jasnow and Kawasaki).     

Large <Emphasis Type="Italic">n</Emphasis> Limit of Gaussian Random Matrices with External Source,Part III: Double Scaling Limit

We consider the double scaling limit in the random matrix ensemble with an external source

Super-Diffusivity in a Shear Flow Model¶from Perpetual Homogenization

This paper is concerned with the asymptotic behavior solutions of stochastic differential equations dy _t =dω _t −∇Γ(y _t ) dt, y ₀=0 and d=2. Γ is a 2 &\times; 2 skew-symmetric matrix associated to a shear flow characterized by an infinite number of spatial scales Γ₁₂=−Γ₂₁=h(x ₁), with h(x ₁)=∑ _{n =0} ^∞γ _n h ⁿ (x ₁/R _n ), where h ⁿ are smooth functions of period 1, h ⁿ (0)=0, γ _n and R _n grow exponentially fast with n. We can show that y ^t has an anomalous fast behavior (?[|y ^t |²]∼t ^1+ν with ν > 0) and obtain quantitative estimates on the anomaly using and developing the tools of homogenization. Received: 1 June 2001 / Accepted: 11 January 2002     

Lattice Dislocations in a 1-Dimensional Model

The spectral properties of the Schr?dinger operator T(t)=−d ²/dx ²+q(x,t) in L ²(ℝ) are studied, where the potential q is defined by q=p(x+t), x>0, and q=p(x), x<0; p is a 1-periodic potential and t∈ℝ is the dislocation parameter. For each t the absolutely continuous spectrum σ _ac (T(t))=σ _ac (T(0)) consists of intervals, which are separated by the gaps γ _n (T(t))=γ _n (T(0))=(α _n ⁻,α _n ⁺), n≥1. We prove: in each gap γ _n ≠?, n≥ 1 there exist two unique “states” (an eigenvalue and a resonance) λ _n ^±(t) of the dislocation operator, such that λ _n ^±(0)=α _n ^± and the point λ _n ^±(t) runs clockwise around the gap γ _n changing the energy sheet whenever it hits α _n ^±, making n/2 complete revolutions in unit time. On the first sheet λ _n ^±(t) is an eigenvalue and on the second sheet λ _n ^±(t) is a resonance. In general, these motions are not monotonic. There exists a unique state λ₀(t) in the basic gap γ₀(T(t))=γ₀(T(0))=(−∞ ,α₀ ⁺). The asymptotics of λ _n ^±(t) as n→∞ is determined. Received: 5 April 1999 / Accepted: 3 March 2000     

The Mixing Time Evolution of Glauber Dynamics for the Mean-Field Ising Model

We consider Glauber dynamics for the Ising model on the complete graph on n vertices, known as the Curie-Weiss model. It is well-known that the mixing-time in the high temperature regime (β < 1) has order n log n, whereas the mixing-time in the case β > 1 is exponential in n. Recently, Levin, Luczak and Peres proved that for any fixed β < 1 there is cutoff at time with a window of order n, whereas the mixing-time at the critical temperature β = 1 is Θ(n ^3/2). It is natural to ask how the mixing-time transitions from Θ(n log n) to Θ(n ^3/2) and finally to exp (Θ(n)). That is, how does the mixing-time behave when β = β(n) is allowed to tend to 1 as n → ∞. In this work, we obtain a complete characterization of the mixing-time of the dynamics as a function of the temperature, as it approaches its critical point β _c  = 1. In particular, we find a scaling window of order around the critical temperature. In the high temperature regime, β = 1 − δ for some 0 < δ < 1 so that δ ² n → ∞ with n, the mixing-time has order (n/δ) log(δ ² n), and exhibits cutoff with constant and window size n/δ. In the critical window, β = 1± δ, where δ ² n is O(1), there is no cutoff, and the mixing-time has order n ^3/2. At low temperature, β = 1 + δ for δ > 0 with δ ² n → ∞ and δ = o(1), there is no cutoff, and the mixing time has order . Research of J. Ding and Y. Peres was supported in part by NSF grant DMS-0605166.     

Center Manifold for Nonintegrable Nonlinear Schrödinger Equations on the Line

In this paper we study the following nonlinear Schr?dinger equation on the line,

Constraints from the old quasar APM 08279+5255 on two classes of Λ(<Emphasis Type="Italic">t</Emphasis>)-cosmologies

The viability of two different classes of Λ(t)CDM cosmologies is tested by using the APM 08279+5255, an old quasar at redshift z = 3.91. In the first class of models, the cosmological term scales as Λ(t) ~ R ⁻ⁿ . The particular case n = 0 describes the standard ΛCDM model whereas n = 2 stands for the Chen and Wu model. For an estimated age of 2 Gyr, it is found that the power index has a lower limit n > 0.21, whereas for 3 Gyr the limit is n > 0.6. Since n can not be so large as ~ 0.81, the ΛCDM and Chen and Wu models are also ruled out by this analysis. The second class of models is the one recently proposed by Wang and Meng which describes several Λ(t)CDM cosmologies discussed in the literature. By assuming that the true age is 2 Gyr it is found that the ε parameter satisfies the lower bound , while for 3 Gyr, a lower limit of is obtained. Such limits are slightly modified when the baryonic component is included.     

Time Quasi-Periodic Unbounded Perturbations¶of Schrödinger Operators and KAM Methods

We eliminate by KAM methods the time dependence in a class of linear differential equations in ℓ² subject to an unbounded, quasi-periodic forcing. This entails the pure-point nature of the Floquet spectrum of the operator H ₀+εP(ωt) for ε small. Here H ₀ is the one-dimensional Schr?dinger operator p ²+V, V(x)∼|x|^α, α <2 for |x|→∞, the time quasi-periodic perturbation P may grow as |x|^β, β <(α−2)/2, and the frequency vector ω is non resonant. The proof extends to infinite dimensional spaces the result valid for quasiperiodically forced linear differential equations and is based on Kuksin's estimate of solutions of homological equations with non-constant coefficients. Received: 3 October 2000 / Accepted: 20 December 2000     

Maximum Solutions of Normalized Ricci Flow on 4-Manifolds

We consider the maximum solution g(t), t ∈ [0,  + ∞), to the normalized Ricci flow. Among other things, we prove that, if (M, ω) is a smooth compact symplectic 4-manifold such that and let g(t), t ∈ [0, ∞), be a solution to (1.3) on M whose Ricci curvature satisfies that |Ric(g(t))| ≤ 3 and additionally χ(M) = 3τ (M) > 0, then there exists an , and a sequence of points {x _j,k ∈ M}, j = 1, . . . , m, satisfying that, by passing to a subsequence,

Convergence to equilibrium distribution. The Klein-Gordon equation coupled to a particle

The Hamiltonian system formed by a Klein-Gordon vector field and a particle in ℝ³ is considered. The initial data of the system are given by a random function, with finite mean energy density, which also satisfies a Rosenblatt- or Ibragimov-type mixing condition. Moreover, initial correlation functions are assumed to be translation invariant. The distribution μ _t of the solution at time t ∈ ℝ is studied. The main result is the convergence of μ _t to a Gaussian measure as t → ∞, where μ_∞ is translation invariant.     

Search for resonant absorption of solar axions emitted in M1 transition in <Superscript>57</Superscript>Fe nuclei

A search for resonant absorption of 14.4 keV solar axions by a ⁵⁷Fe target was performed. The Si(Li) detector placed inside the low-background setup was used to detect the γ-quanta appearing in the deexcitation of the 14.4 keV nuclear level: A+⁵⁷Fe→⁵⁷Fe^*→⁵⁷Fe+γ. The new upper limit for the hadronic axion mass has been obtained of m _A ≤159 eV (95% c.l.) (S=0.5, z=0.56).     

Symmetry Transitions in Random Matrix Theory &; <Emphasis Type="Italic">L</Emphasis>-functions

We compute the moments of the characteristic polynomials of random orthogonal and symplectic matrices, defined by averages with respect to Haar measure on SO(2N) and USp(2N), to leading order as N → ∞, on the unit circle as functions of the angle θ measured from one of the two symmetry points in the eigenvalue spectrum . Our results extend previous formulae that relate just to the symmetry points, i.e. to θ = 0. Local spectral statistics are expected to converge to those of random unitary matrices in the limit as N → ∞ when θ is fixed, and to show a transition from the orthogonal or symplectic to the unitary forms on the scale of the mean eigenvalue spacing: if θ = π y/N they become functions of y in the limit when N → ∞. We verify that this is true for the spectral two-point correlation function, but show that it is not true for the moments of the characteristic polynomials, for which the leading order asymptotic approximation is a function of θ rather than y. Symmetry points therefore influence the moments asymptotically far away on the scale of the mean eigenvalue spacing. We also investigate the moments of the logarithms of the characteristic polynomials in the same context. The moments of the characteristic polynomials of random matrices are conjectured to be related to the moments of families of L-functions. Previously, moments at the symmetry point θ = 0 have been related to the moments of families of L-functions evaluated at the centre of the critical strip. Our results motivate general conjectures for the moments of orthogonal and symplectic families of L-functions evaluated at a fixed height t up the critical line. These conjectures suggest that the symmetry of the non-trivial zeros of the L-functions influences the moments asymptotically far, on the scale of the mean zero spacing, from the centre of the critical strip. We verify that the second moments of real quadratic Dirichlet L-functions and a family of automorphic L-functions are consistent with our conjectures. JPK is supported by an EPSRC Senior Research Fellowship. BEO was supported by an Overseas Research Scholarship and a University of Bristol Research Scholarship.     

Wetting Transition on a One-Dimensional Disorder

We consider wetting of a one-dimensional random walk on a half-line x≥0 in a short-ranged potential located at the origin x=0. We demonstrate explicitly how the presence of a quenched chemical disorder affects the pinning-depinning transition point. For small disorders we develop a perturbative technique which enables us to compute explicitly the averaged temperature (energy) of the pinning transition. For strong disorder we compute the transition point both numerically and using the renormalization group approach. Our consideration is based on the following idea: the random potential can be viewed as a periodic potential with the period n in the limit n→∞. The advantage of our approach stems from the ability to integrate exactly over all spatial degrees of freedoms in the model and to reduce the initial problem to the analysis of eigenvalues and eigenfunctions of some special non-Hermitian random matrix with disorder-dependent diagonal and constant off-diagonal coefficients. We show that even for strong disorder the shift of the averaged pinning point of the random walk in the ensemble of random realizations of substrate disorder is indistinguishable from the pinning point of the system with preaveraged (i.e. annealed) Boltzmann weight.     

Convergent polynomial expansion,energy dependence of slope parameters,scaling hypothesis and predictions forπ ± p andK + p scattering

A phenomenological representation for differential cross-section recently proposed using Mandelstam analyticity and convergent polynomial expansion (CPE) which has been found to be successful in describing scaling of the differential cross-section-ratio data for several elastic diffractive and inelastic nondiffractive processes is used to analyse the energy dependence of the slope-parameter data at high energies, extrapolate the slope parameter and predict the differential cross-section ratio as a function of |t| at higher energies forπ ^± pndK ⁺ p scattering. Following the method of Hansen and Krisch it is found that, in spite of the existence of rather widely varying data points for nearbys values, a more systematic trend in the energy dependence of the slope parameter emerges when a statistical average of the existing high-energy data is used. Extrapolating the fits to the average data ontos → ∞ provides strong evidence in favour of a model-independent result that asymptotically theπ ^± p slopes may be equal. There is also a strong indication to the effect that each of these two slopes may be equal to theK ⁺ p slope fors → ∞. Using the scaling curves generated by the existing data on differential cross-section ratio and extrapolated values of the slope parameter, the differential cross-section ratio for each of the three processes is predicted as a function of |t| for higher energies.     

Burning Cars in a Parking Lot

Knuth’s parking scheme is a model in computer science for hashing with linear probing. One may imagine a circular parking lot with n sites; cars arrive at each site with unit rate. When a car arrives at a vacant site, it parks there; otherwise it turns clockwise and parks at the first vacant site which is found. We incorporate fires into this model by throwing Molotov cocktails on each site at a smaller rate n ^−α , where 0 < α < 1 is a fixed parameter. When a car is hit by a Molotov cocktail, it burns and the fire propagates to the entire occupied interval which turns vacant. We show that with high probability when n → ∞, the parking lot becomes saturated at a time close to 1 (i.e. as in the absence of fire) for α > 2/3, whereas for α < 2/3, the average occupation approaches 1 at time 1 but then quickly drops to 0 before the parking lot is ever saturated. Our study relies on asymptotics for the occupation of the parking lot without fires in certain regimes which may be of independent interest.