共查询到20条相似文献,搜索用时 156 毫秒
1.
A. B. J. Kuijlaars A. Martínez-Finkelshtein F. Wielonsky 《Communications in Mathematical Physics》2011,308(1):227-279
We consider the double scaling limit for 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 = 1 at x = 0. After appropriate rescaling, the paths fill a region in the tx–plane as n → ∞ that intersects the hard edge at x = 0 at a critical time t = t
*. In a previous paper, the scaling limits for the positions of the paths at time t ≠ t
* were shown to be the usual scaling limits from random matrix theory. Here, we describe the limit as n → ∞ of the correlation kernel at critical time t
* and in the double scaling regime. We derive an integral representation for the limit kernel which bears some connections
with the Pearcey kernel. The analysis is based on the study of a 3 × 3 matrix valued Riemann-Hilbert problem by the Deift-Zhou
steepest descent method. The main ingredient is the construction of a local parametrix at the origin, out of the solutions
of a particular third-order linear differential equation, and its matching with a global parametrix. 相似文献
2.
Steven Delvaux 《Communications in Mathematical Physics》2013,324(3):715-766
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. 相似文献
3.
Anne Boutet de Monvel Athanassis S. Fokas Dmitry Shepelsky 《Communications in Mathematical Physics》2006,263(1):133-172
Let q(x,t) satisfy an integrable nonlinear evolution PDE on the interval 0<x<L, and let the order of the highest x-derivative be n. For a problem to be at least linearly well-posed one must prescribe N boundary conditions at x=0 and n−N boundary conditions at x=L, where if n is even, N=n/2, and if n is odd, N is either (n−1)/2 or (n+1)/2, depending on the sign of ∂nxq. For example, for the sine-Gordon (sG) equation one must prescribe one boundary condition at each end, while for the modified
Korteweg-de Vries (mKdV) equations involving qt+qxxx and qt−qxxx one must prescribe one and two boundary conditions, respectively, at x=0. We will refer to these two mKdV equations as mKdV-I and mKdV-II, respectively.
Here we analyze the Dirichlet problem for the sG equation, as well as typical boundary value problems for the mKdV-I and mKdV-II
equations. We first show that the unknown boundary values at each end (for example, qx(0,t) and qx(L,t) in the case of the Dirichlet problem for the sG equation) can be expressed in terms of the given initial and boundary conditions
through a system of four nonlinear ODEs. We then show that q(x,t) can be expressed in terms of the solution of a 2×2 matrix Riemann-Hilbert problem formulated in the complex k-plane. This problem has explicit (x,t) dependence in the form of an exponential; for example, for the case of the sG this exponential is exp {i(k−1/k)x+i(k+1/k)t}. Furthermore, the relevant jump matrices are explicitly given in terms of the spectral functions {a(k),b(k)}, {A(k),B(k)}, and , which in turn are defined in terms of the initial conditions, of the boundary values of q and of its x-derivatives at x=0, and of the boundary values of q and of its x-derivatives at x=L, respectively. This Riemann-Hilbert problem has a global solution. 相似文献
4.
J. F. Jesus 《General Relativity and Gravitation》2008,40(4):791-798
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. 相似文献
5.
Wen-Xin Qin 《Communications in Mathematical Physics》2012,311(2):513-538
In this paper we are mainly concerned with existence and modulation of uniform sliding states for particle chains with damping
γ and external driving force F. If the on-site potential vanishes, then for each F > 0 there exist trivial uniform sliding states x
n
(t) = n
ω + ν
t + α for which the particles are uniformly spaced with spacing ω > 0, the sliding velocity of each particle is ν = F/γ, and the phase α is arbitrary. If the particle chain with convex interaction potential is placed in a periodic on-site potential,
we show under some conditions the existence of modulated uniform sliding states of the form
xn(t)=nw+nt+a+u(nw+nt+a),x_n(t)=n\omega+\nu t+\alpha+u(n\omega+\nu t+\alpha), 相似文献
6.
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 相似文献
7.
In this paper, we study the singularly perturbed Laguerre unitary ensemble $$\frac{1}{Z_n} ({\rm det}\,\, M)^\alpha e^{- {\rm tr}\, V_t(M)}dM, \qquad \alpha > 0,$$ with \({V_t(x) = x + t/x,\,\, x \in (0,+\infty)}\) and t > 0. Due to the effect of t/x for varying t, the eigenvalue correlation kernel has a new limit instead of the usual Bessel kernel at the hard edge 0. This limiting kernel involves \({\psi}\) -functions associated with a special solution to a new third-order nonlinear differential equation, which is then shown to be equivalent to a particular Painlevé III equation. The transition of this limiting kernel to the Bessel and Airy kernels is also studied when the parameter t changes in a finite interval (0, d]. Our approach is based on Deift–Zhou nonlinear steepest descent method for Riemann–Hilbert problems. 相似文献
8.
Evgeni Korotyaev 《Communications in Mathematical Physics》2000,213(2):471-489
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 相似文献
9.
W. J. Kossler X. H. Yu A. Greer H. E. Schone C. E. Stronach M. Davis R. S. Cary W. F. Lankford A. Moodenbaugh J. Oostens 《Hyperfine Interactions》1991,63(1-4):253-258
Transverse-and zero-field μSR measurements have been made for YBa2(Cu1−x
Fe
x
)3O7 withx=0.04, 0.08 and 0.12. The temperature range studied was from approximately 7.5 K to 100 K. The onset of magnetic ordering
commences at about 7.5 K forx=0.04, 10 K forx=0.08 and 20 K forx=0.12. The Gaussian depolarization parameter, σ ofG
x
(t) = exp(−σ2
t
2/2), is depressed by a factor of about 0.6 forx=0.04, but for thex=0.08 sample σ is depressed by a factor of 10 and increasing suppression is seen as the temperature is lowered below 45 K.
This decrease in σ is interpreted in terms of decreasing electronic mean free paths. 相似文献
10.
The initial value problem for an integrable system, such as the Nonlinear Schrödinger equation, is solved by subjecting the linear eigenvalue problem arising from its Lax pair to inverse scattering, and, thus, transforming it to a matrix Riemann-Hilbert problem (RHP) in the spectral variable. In the semiclassical limit, the method of nonlinear steepest descent ([4,5]), supplemented by the g-function mechanism ([3]), is applied to this RHP to produce explicit asymptotic solution formulae for the integrable system. These formule are based on a hyperelliptic Riemann surface ${\mathcal {R} = \mathcal {R}(x,t)}
11.
For aL
∞
(ℝ+)∩L
1
(ℝ+) the truncated Bessel operator B
τ
(a) is the integral operator acting on L
2
[0,τ] with the kernel
12.
We consider a particle system of the squared Bessel processes with index ν>−1 conditioned never to collide with each other,
in which if −1<ν<0 the origin is assumed to be reflecting. When the number of particles is finite, we prove for any fixed
initial configuration that this noncolliding diffusion process is determinantal in the sense that any multitime correlation
function is given by a determinant with a continuous kernel called the correlation kernel. When the number of particles is
infinite, we give sufficient conditions for initial configurations so that the system is well defined. There the process with
an infinite number of particles is determinantal and the correlation kernel is expressed using an entire function represented
by the Weierstrass canonical product, whose zeros on the positive part of the real axis are given by the particle-positions
in the initial configuration. From the class of infinite-particle initial configurations satisfying our conditions, we report
one example in detail, which is a fixed configuration such that every point of the square of positive zero of the Bessel function
J
ν is occupied by one particle. The process starting from this initial configuration shows a relaxation phenomenon converging
to the stationary process, which is determinantal with the extended Bessel kernel, in the long-term limit. 相似文献
13.
For the asymmetric simple exclusion process η
t
on the integer lattice with two-sided Bernoulli initial condition, we derive exact formulas for the following quantities:
(1) ℙ(η
t
(x)=1), the probability that site x is occupied at time t; (2) the correlation function ℙ(η
t
(x)=1,η
0(0)=1); (3) the distribution function for Q
t
, the total flux across 0 at time t, and its exponential generating function. 相似文献
14.
We obtain an asymptotic expansion for the solution of the Cauchy problem for the Korteweg-de Vries (KdV) equation
15.
Large <Emphasis Type="Italic">n</Emphasis> Limit of Gaussian Random Matrices with External Source,Part III: Double Scaling Limit 总被引:1,自引:1,他引:0
We consider the double scaling limit in the random matrix ensemble with an external source
16.
Manuela Girotti 《Mathematical Physics, Analysis and Geometry》2014,17(1-2):183-211
We consider the gap probability for the Generalized Bessel process in the single-time and multi-time case, a determinantal process which arises as critical limiting kernel in the study of self-avoiding squared Bessel paths. We prove that such gap probabilities, i.e. the scalar and matrix Fredholm determinant of the process respectively, can be expressed in terms of determinants of suitable Its-Izergin-Korepin-Slavnov integrable kernels and therefore they can be related in a canonical way to Riemann-Hilbert problems. Moreover, such Fredholm determinants can be interpreted as isomonodromic τ-functions in the sense of Jimbo, Miwa and Ueno; in particular, in the single-time case we are able to construct an explicit Lax pair. On the other hand, in the multi-time case we explicitly define a completely new multi-time kernel and we proceed with the study of gap probabilities as in the single-time case. 相似文献
17.
We consider the random matrix ensemble with an external sourcedefined on n×n Hermitian matrices, where A is a diagonal matrix with only two eigenvalues ±a of equal multiplicity. For the case a>1, we establish the universal behavior of local eigenvalue correlations in the limit n, which is known from unitarily invariant random matrix models. Thus, local eigenvalue correlations are expressed in terms of the sine kernel in the bulk and in terms of the Airy kernel at the edge of the spectrum. We use a characterization of the associated multiple Hermite polynomials by a 3×3-matrix Riemann-Hilbert problem, and the Deift/Zhou steepest descent method to analyze the Riemann-Hilbert problem in the large n limit.Dedicated to Freeman Dyson on his eightieth birthdayThe first author was supported in part by NSF Grants DMS-9970625 and DMS-0354962.The second author was supported in part by projects G.0176.02 and G.0455.04 of FWO-Flanders, by K.U.Leuven research grant OT/04/24, and by INTAS Research Network NeCCA 03-51-6637. 相似文献
18.
Eunghyun Lee 《Journal of statistical physics》2010,140(4):635-647
In this paper we give the distribution of the position of a particle in the asymmetric simple exclusion process (ASEP) with
the alternating initial condition. That is, we find ℙ(X
m
(t)≤x) where X
m
(t) is the position of the particle at time t which was at m=2k−1, k∈ℤ at t=0. As in the ASEP with step initial condition, there arises a new combinatorial identity for the alternating initial condition,
and this identity relates the integrand of the integral formula for ℙ(X
m
(t)≤x) to a determinantal form together with an extra product. 相似文献
19.
We characterize averages of ?l=1N|x - tl|a- 1{\prod_{l=1}^N|x - t_l|^{\alpha - 1}} with respect to the Selberg density, further constrained so that tl ? [0,x] (l=1,...,q){t_l \in [0,x] (l=1,\dots,q)} and tl ? [x,1] (l=q+1,...,N){t_l \in [x,1] (l=q+1,\dots,N)} , in terms of a basis of solutions of a particular Fuchsian matrix differential equation. By making use of the Dotsenko-Fateev
integrals, the explicit form of the connection matrix from the Frobenius type power series basis to this basis is calculated,
thus allowing us to explicitly compute coefficients in the power series expansion of the averages. From these we are able
to compute power series for the marginal distributions of the tj (j=1,...,N){t_j (j=1,\dots,N)} . In the case q = 0 and α < 1 we compute the explicit leading order term in the x ? 0{x \to 0} asymptotic expansion, which is of interest to the study of an effect known as singularity dominated strong fluctuations.
In the case q = 0 and
a ? \mathbbZ+{\alpha \in \mathbb{Z}^+} , and with the absolute values removed, the average is a polynomial, and we demonstrate that its zeros are highly structured. 相似文献
20.
In this paper, we study the asymptotic behavior of solutions of semilinear abstract differential equations (*) u′(t) = Au(t) + t
n
f(t, u(t)), where A is the generator of a C
0-semigroup (or group) T(·), f(·, x) ∈ A for each x ∈ X, A is the class of almost periodic, almost automorphic or Levitan almost periodic Banach space valued functions ϕ: ℝ → X and n ∈ {0, 1, 2, ...}. We investigate the linear case when T(·)x is almost periodic for each x ∈ X; and the semilinear case when T(·) is an asymptotically stable C
0-semigroup, n = 0 and f(·, x) satisfies a Lipschitz condition. Also, in the linear case, we investigate (*) when ϕ belongs to a Stepanov class S
p-A defined similarly to the case of S
p-almost periodic functions. Under certain conditions, we show that the solutions of (*) belong to A
u:= A ∩ BUC(ℝ, X) if n = 0 and to t
n
A
u ⊕ w
n
C
0 (ℝ, X) if n ∈ ℕ, where w
n(t) = (1 + |t|)n. The results are new for the case n ∈ ℕ and extend many recent ones in the case n = 0.
Dedicated to the memory of B. M. Levitan 相似文献
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