# One-Dimensional Coulomb Multiparticle Systems

Authors: V. A. Malyshev; A. A. Zamyatin
Journal: Advances in Mathematical Physics
 (2015)

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## Abstract

We consider the system of particles with equal charges and nearest neighbour Coulomb interaction on the interval. We study local properties of this system, in particular the distribution of distances between neighbouring charges. For zero temperature case there is sufficiently complete picture and we give a short review. For Gibbs distribution the situation is more difficult and we present two related results.


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## Body

## 1. Introduction

Many electric phenomena are not well understood and even might seem mysterious. More exactly, most are not still deduced from the microscale version of Maxwell equations on rigorous mathematical level. For example, even in the standard direct or alternative current, the electrons move along hundred kilometers of power lines, but the external accelerating force acts only on some meters of the wire. Here what one can read about this in “The Feynman Lectures on Physics” (see [1], volume 2, 16-2):“…The force pushes the electrons along the wire. But why does this move the galvanometer, which is so far from the force? Because when the electrons which feel the magnetic force try to move, they push - by electric repulsion - the electrons a little farther down the wire; they, in turn, repel the electrons a little farther on, and so on for a long distance. An amazing thing. It was so amazing to Gauss and Weber - who first built a galvanometer - that they tried to see how far the forces in the wire would go. They strung the wire all the way across the city….”This was written by the famous physicist Richard Feynman. However, after that, this “amazing thing” was vastly ignored in the literature. For example, the Drude model (that can be found in any textbook on solid state physics, see, e.g., [2]), considers free (noninteracting) electrons and constant external accelerating force acting along all the wire, without mentioning where this force (or field) comes from.Many more questions arise. For example, why the electrons in DC move slowly, but such stationary regime is being established almost immediately. In [3, 4] it was demonstrated rigorously that even on the classical (not quantum) level that the stationary and space homogeneous flow of charged particles may exist as a result of self-organization of strongly interacting (via Coulomb repulsion) system of electrons. This means that the field accelerating the electrons is created by the neighboring electrons via some multiscale self-organization.In fact, Ohm’s law is formulated on the macroscale (of order one); one-dimensional movement ofN electrons is described on the microscale (of order N-1), but the accelerating force is the corollary of the processes on the so-called submicroscale (of the order N-2). To show this we used not only classical nonrelativistic physics, Newtonian dynamics and Coulomb’s law, but also the simplest friction mechanism, ignoring where this friction mechanism comes from.Besides other nonsolved dynamic problems like discharge, lightning, global current, bioelectricity, and so forth, also local and global properties of equilibrium configurations of charged particles in external electric fields are not at all studied (note that the mathematical part of equilibrium statistical physics has been developed mostly on the lattice). The equilibrium configurations can be either ground state (zero temperature) or Gibbs states. Ground states are easier to describe and we give short review of results in Section2 of the paper. Study of local structure of Gibbs configurations is at the very beginning and we present in Section 3 two new results with complete proofs.


## 2.
Part I: Ground State Configurations

Consider systems ofN particles with equal charges, Coulomb interaction and external force F on an manifold. Even when there is no external force, the problem appears to be sufficiently difficult and was claimed important already long ago [5]. For example, J. J. Thomson (who discovered electron) suggested the problem of finding such configurations on the sphere, and the answer has been known for N = 2, 3, and 4 for more than 100 years, but for N=5 the solution was obtained only quite recently [6].More interesting is the case of largeN, where the asymptotics N→∞ is of main interest. In one-dimensional case T. J. Stieltjes studied the problem with logarithmic interaction and found its connection with zeros of orthogonal polynomials on the corresponding interval (see [7, 8]). However, the problem of finding minimal energy configurations on two-dimensional sphere for any N and power interaction (sometimes it is called the seventh problem of Smale; it is also connected with the names of F. Risz and M. Fekete) was completely solved only for quadratic interaction (see [9–11] and review [12]). For more general compact manifolds see review [13].In this section we review recent results concerning nonzero external force. Moreover, we consider not only global minima but even more interesting case of local energy minima. It appears that even in the simpler one-dimensional model with nearest neighbour interaction there is an interesting structure of fixed points (more exactly, fixed configurations), rich both in the number and in the charge distribution.


### 2.1. The Model

We consider the set of configurations ofN+1 point particles:(1)-L≤xN<⋯<x1<x0≤0with equal charges on the segment [-L,0]. Here N is assumed to be sufficiently large. We assume repulsive Coulomb interaction of nearest neighbours and external force αextF0(x); that is, the potential energy is given by(2)U=∑i=1NVxi-1-xi-∑i=0N∫-LxiαextF0xdx,Vx=αintx,where αext, αint are positive constants. This defines the dynamics of the system of charges, if one defines exactly what occurs with particles 0 and N in the points 0 and -L correspondingly. Namely, we assume completely inelastic boundary conditions. More exactly, when particle x0(t) at time t reaches point 0, having some velocity v0(t-0)≥0, then its velocity v0(t) immediately becomes zero, and the particle itself stays at point 0 until the force acting on it (which varies accordingly to the motion of other particles) becomes negative. The same occurs for the particle xN(t) at point -L.


### 2.2. Problem of Many Local Minima

It is evident that ifF0≡0, then there is only one fixed point with(3)δk=xk-1-xk=LN,k=1,…,N.Thus it is the global energy minimum. More general result is the following.Theorem 1.
Assume thatF0(x) is continuous, nonnegative, and monotonic. Then for any N, L, and αren the fixed point exists and is unique.However, the monotonicity assumption in this theorem is very essential. An example of strong nonuniqueness (where the number of fixed points is of the orderN) is very simple—for a function F0(x) with the only maximum inside the interval. Namely, on the interval [-1,1], put for b>a>0(4)F0x=a-2ax,x≥0,F0x=a+2bx,x≤0.Then there exists Ccr>0 such that for all sufficiently large N and αren=cN,c>Ccr, one can show using similar techniques that for any odd N1<N there exists fixed point such that(5)-1=xN<⋯<xN1<0<xN1-1<⋯<xN1+1/2=12<⋯<x0<1.Moreover, any such point will give local minimum of the energy.One-dimensional case shows what can be expected in multidimensional case, which is more complicated but has great interest in connection to the static charge distribution in the atmosphere or in the living organism.


### 2.3. Phase Transitions

To discover phase transitions one should consider asymptoticsN→∞, with the parameters L,F0(x) being fixed. Then the fixed points will depend only on the “renormalized force” F=(αext/αint)Fo, and we assume that the renormalized constant αren=αext/αint can tend to infinity together with N, namely, as αren=cNγ, where c,γ>0.The necessity to consider cases whenαren depends on N issues from concrete examples, where αren≫N. For example, the linear density of electrons in some conductors, see [2], is of the order N≈109 m−1, αint=e2/ϵ0≈10-28, and αext=220 (volt/meter) where e=220×10-19 (in SI system). Thus αren has the order 1011. This is close to the critical point of our model, which, as it will be shown, is asymptotically ccrN that is close to 4×109 in our case.Below this section we assume for simplicity thatF0>0 is constant. We formulate now the assertions proven in [14–16].Critical Force. For any N,L there exists Fcr=Fcr(N,L) such that for the fixed point the following holds: xN>-L for F>Fcr and xN=-L for F≤Fcr. If F=cNγ,γ>1, then for any c>0 we have xN→0. At the same time Fcr~N→∞ccrN, where(6)ccr=4L2.Multiscale Phase. The case when αren does not depend on N was discussed in detail in [14, 15]; there are no phase transitions, but it is discovered that the structure of the fixed configuration differs from (3) only on the submicroscale of the order N-2. More exactly, consider more general case when V(x)=|x|-b,b>0. Then the following holds: if F does not depend on N, then for any k=1,…,N(7)xk-1-xk-LN~FL1+b1+bN-bk-N2.Uniform Density. We define the density ρ(x) so that for any subintervals I⊂[-L,0] there exist the limits(8)ρI=∫Iρxdx=limN→∞⁡#i:xi∈IN.Then if F=o(N), then the density exists and is strictly uniform, that is, for all k=1,…,N as N→∞,(9)maxk⁡xk-1-xk-LN=o1N.Nonuniform Density. If F=cN and 0<c≤ccr, then xN=-L and the density of particles exists and is nowhere zero but is not uniform (not constant in x).Weak Contraction. If F=cN and c>ccr, then, as N→∞,(10)-L<xN⟶-2cand the density on the interval (-2/c,0) is not uniform.Strong Contraction. If F=cNγ,γ>1, then the density ρ(x)→δ(x) in the sense of distributions.Both contraction cases are related to the discharge possibility; as after disappearance of the external force, discharge can be produced, the strength of which depends on the initial concentration of charged particles.


## 2.1. The Model

We consider the set of configurations ofN+1 point particles:(1)-L≤xN<⋯<x1<x0≤0with equal charges on the segment [-L,0]. Here N is assumed to be sufficiently large. We assume repulsive Coulomb interaction of nearest neighbours and external force αextF0(x); that is, the potential energy is given by(2)U=∑i=1NVxi-1-xi-∑i=0N∫-LxiαextF0xdx,Vx=αintx,where αext, αint are positive constants. This defines the dynamics of the system of charges, if one defines exactly what occurs with particles 0 and N in the points 0 and -L correspondingly. Namely, we assume completely inelastic boundary conditions. More exactly, when particle x0(t) at time t reaches point 0, having some velocity v0(t-0)≥0, then its velocity v0(t) immediately becomes zero, and the particle itself stays at point 0 until the force acting on it (which varies accordingly to the motion of other particles) becomes negative. The same occurs for the particle xN(t) at point -L.


## 2.2. Problem of Many Local Minima

It is evident that ifF0≡0, then there is only one fixed point with(3)δk=xk-1-xk=LN,k=1,…,N.Thus it is the global energy minimum. More general result is the following.Theorem 1.
Assume thatF0(x) is continuous, nonnegative, and monotonic. Then for any N, L, and αren the fixed point exists and is unique.However, the monotonicity assumption in this theorem is very essential. An example of strong nonuniqueness (where the number of fixed points is of the orderN) is very simple—for a function F0(x) with the only maximum inside the interval. Namely, on the interval [-1,1], put for b>a>0(4)F0x=a-2ax,x≥0,F0x=a+2bx,x≤0.Then there exists Ccr>0 such that for all sufficiently large N and αren=cN,c>Ccr, one can show using similar techniques that for any odd N1<N there exists fixed point such that(5)-1=xN<⋯<xN1<0<xN1-1<⋯<xN1+1/2=12<⋯<x0<1.Moreover, any such point will give local minimum of the energy.One-dimensional case shows what can be expected in multidimensional case, which is more complicated but has great interest in connection to the static charge distribution in the atmosphere or in the living organism.


## 2.3. Phase Transitions

To discover phase transitions one should consider asymptoticsN→∞, with the parameters L,F0(x) being fixed. Then the fixed points will depend only on the “renormalized force” F=(αext/αint)Fo, and we assume that the renormalized constant αren=αext/αint can tend to infinity together with N, namely, as αren=cNγ, where c,γ>0.The necessity to consider cases whenαren depends on N issues from concrete examples, where αren≫N. For example, the linear density of electrons in some conductors, see [2], is of the order N≈109 m−1, αint=e2/ϵ0≈10-28, and αext=220 (volt/meter) where e=220×10-19 (in SI system). Thus αren has the order 1011. This is close to the critical point of our model, which, as it will be shown, is asymptotically ccrN that is close to 4×109 in our case.Below this section we assume for simplicity thatF0>0 is constant. We formulate now the assertions proven in [14–16].Critical Force. For any N,L there exists Fcr=Fcr(N,L) such that for the fixed point the following holds: xN>-L for F>Fcr and xN=-L for F≤Fcr. If F=cNγ,γ>1, then for any c>0 we have xN→0. At the same time Fcr~N→∞ccrN, where(6)ccr=4L2.Multiscale Phase. The case when αren does not depend on N was discussed in detail in [14, 15]; there are no phase transitions, but it is discovered that the structure of the fixed configuration differs from (3) only on the submicroscale of the order N-2. More exactly, consider more general case when V(x)=|x|-b,b>0. Then the following holds: if F does not depend on N, then for any k=1,…,N(7)xk-1-xk-LN~FL1+b1+bN-bk-N2.Uniform Density. We define the density ρ(x) so that for any subintervals I⊂[-L,0] there exist the limits(8)ρI=∫Iρxdx=limN→∞⁡#i:xi∈IN.Then if F=o(N), then the density exists and is strictly uniform, that is, for all k=1,…,N as N→∞,(9)maxk⁡xk-1-xk-LN=o1N.Nonuniform Density. If F=cN and 0<c≤ccr, then xN=-L and the density of particles exists and is nowhere zero but is not uniform (not constant in x).Weak Contraction. If F=cN and c>ccr, then, as N→∞,(10)-L<xN⟶-2cand the density on the interval (-2/c,0) is not uniform.Strong Contraction. If F=cNγ,γ>1, then the density ρ(x)→δ(x) in the sense of distributions.Both contraction cases are related to the discharge possibility; as after disappearance of the external force, discharge can be produced, the strength of which depends on the initial concentration of charged particles.


## 3.
Part II: Gibbs Distribution

We consider the setΩ=ΩN={ω=(x0,…,xN)},N≥2, of configurations of N+1 points particles on the segment [0,L] such that(11)0=x0<⋯<xN=L.Introducing new variables uk=xk-xk-1,k=1,…,N, one sees that ΩN is an open simplex:(12)u1+⋯+uN=L,uk>0,which is denoted by S(N,L).We will consider the probability density:(13)Pω=ZN-1exp⁡-βUωon S(N,L) with respect to the Lebesgue measure ν on S(N,L), where(14)ZN=∫SN,Lexp⁡-βUωdν=∫0<x1<⋯<xN-1<Lexp⁡-βUωdx1⋯dxN-1=∫SN,L∏k=1Nexp⁡-βVukdu1⋯duN,Uω=Vx1-x0+⋯+VxN-xN-1=∑k=1NVukis the function on the set of configurations called the energy and V is the function on the segment [0,∞], called potential. The most interesting case for us is the Coulomb repulsive potential:(15)Vu=1u,u>0.Equivalently one could say that we consider the sum(16)SN=ξ1+⋯+ξNof N independent identically distributed positive random variables ξk, each having density g(u),u>0, further assumed to be smooth, for simplicity. Then the conditional density of the vector {ξ1,…,ξN}, under the condition SN=L,(17)Pω=gu1⋯guN∫SN,Lgu1⋯guNdu1⋯duNthat coincides with (13), if we put(18)gu=Z1-1exp⁡-βVu,Z1=∫0Lexp⁡-βVudu.It is clear that conditional distributions P(ξk<x∣SN=L) are the same for all k. In particular(19)ξk∣SN=L=LN.Note that in the limiting case β=∞ the distribution is concentrated in the unique fixed point uk=L/N.Below we putL=1. Let(20)f⋆nx=f⋆f⋆f⋆⋯⋆fbe the n times convolution of f(x). Then the conditional variance is(21)dN=Dξ1∣SN=1=∫01x-1N2gxg⋆N-11-xg⋆N1dx.We want to note here that there exist many papers, related to the famous Kac mean field model, where conditional independence (chaos) ofξk is proved under various conditions (see, e.g., [17] and the references therein). We follow here another goal, that is, revealing possible multiscale local structure in the Gibbs situation, which could resemble zero temperature case structure, discussed in Section 2.


### 3.1. Results

We consider the densities having the following asymptotic behaviour asx→0:(22)gx~c0xα-1e-β/x,α∈R,β≥0.Theorem 2.
Under this condition and ifα>0 and β=0, as N→∞,(23)dN~c1N-2for some constant c1=c1(α)>0 depending only on α.Theorem 3.
Ifβ>0, then for any α∈R, as N→∞,(24)dN~c1N-3for some constant c1=c1(β)>0 depending only on β.It is of interest to know the behaviour of the covariance for densities with hyperexponential decrease at zero.


### 3.2. Proofs

#### 3.2.1. General Power Asymptotics

We will prove here Theorem2. Instead of one density g(x) it is useful to consider the family of densities (such trick has been used in some large deviations problems, see [18, 19])(25)hλx=e-λxgxz-1λ,where λ≥0 and(26)zλ=∫01e-λxgxdx.Let ξλ,k be random variables with density hλ(x). Put mλ=Eξλ,k, σλ2=Dξλ,k and denote the conditional densities of ξλ,k under the condition that SN=1:(27)fλx=hλxhλ⋆N-11-xhλ⋆N1=gxg⋆n-11-xg⋆n1,x∈0,1.It is easy to check that fλ(x) in fact does not depend on λ.Lemma 4.
(1)mλ~αλ-1 and σλ2~αλ-2 as λ→∞. (2) there exists a unique λN, such that mλN=1/N. Then λN~αN, σλn2~α-1N-2 as N→∞.Proof.
(1) By abelian theorem, see [20, page 445, Theorem  3], and the condition g(x)~c0xα-1 as x→0 we have as λ→∞(28)zλ~Γαc0λN,-z′λ~Γα+1c0λN+1,z′′λ~Γα+2c0λN+2.It follows that(29)mλ=-z′λzλ~Γα+1Γαλ=αλ,σλ2=c′′λcλ-c′λcλ2~Γα+2Γα-Γα+1Γα2λ-2=α2λ2as λ→∞.
(2) The functionmλ is monotonically decreasing in λ. Thus for any N there exists λN such that(30)mλN=1N.From (1) it follows that λN~αN and σλN2~α-1N-2 as N→∞.Letϕλ(t) be the characteristic function of ξλ.Lemma 5.
The family of densities (25) has the following properties: (1)
The normalized momentaλ=E|ξλ-mλ|4/σλ4 is bounded uniformly in λ>0.(2)
For anyδ>0 there exists λ0>0 such that supλ>λ0supt>δϕλ(t/σλ)<1.(3)
For someq≥1(31)∫-∞∞ϕλtqdt=Oλqα.Proof.
(1) It is similar to the proof of (1) in Lemma4.
(2) Put(32)ft,λ=ϕλtσλ-1=z-1λ∫01eitσλ-1xe-λxgxdx.Let us show that for some δ>0(33)ft,λ≤11-itα-1/2α+Oe-δλas λ→∞. For this we can write f(t,λ) as(34)ft,λ=J1λ+J2λ,where(35)J1λ=z-1λ∫0λ-1/2eitσλ-1xe-λxgxdx,J2λ=z-1λ∫λ-1/21eitσλ-1xe-λxgxdx.Taking into account that(36)zλ~c0Γαλ-α,σλ2~αλ-2,λ⟶∞and condition g(x)~c0xα-1 as x→0 we have that as λ→∞(37)J1λ~Γα-1λα∫0λ-1/2eitα-1/2λxe-λxxα-1dx.Putting y=λx in the last integral we get(38)Γα-1λα∫0λ-1/2eitα-1/2λxe-λxxα-1dx=Γα-1∫0λ1/2eitα-1/2ye-yyα-1dy.As (39)Γα-1∫0λ1/2eitα-1/2ye-yyα-1dy=Γα-1∫0∞eitα-1/2ye-yyα-1dy-Γα-1∫λ1/2∞eitα-1/2ye-yyα-1dy,Γα-1∫0∞eitα-1/2ye-yyα-1dy=11-itα-1/2α,Γα-1∫λ1/2∞eitα-1/2ye-yyα-1dy=Oe-δλfor some δ>0, then(40)J1λ≤11-itα-1/2α+Oe-δλ.For J2(λ) we get the estimate(41)J2λ=c-1λ∫λ-1/21eitσλ-1xe-λxgxdx≤Cλαe-λ=Oe-δλ.From these estimates (33) follows.
(3) Note that always(42)∫01hλxpdx<∞for some p>1. Without loss of generality one can assume that 1<p≤2. By Hausdorff-Young inequality(43)∫-∞∞ϕλtqdt1/q≤12π∫01hλxpdx1/p,where 1<p≤2 and 1/p+1/q=1. As(44)12π∫01hλxpdt1/p=z-1λ12π∫01e-λxgxpdt1/p≤z-1λ12π∫01gxpdt1/pand z(λ)~Cλ-α, then(45)∫-∞∞ϕλtqdt=Oλqα.Lemma 6.
Assume conditions (1)–(3) of Lemma5 and that λN are defined by the condition mλN=1/N. Then(46)hλN⋆Nx=1σλN2πNexp⁡-x-122σλN2N+o1,where o(1) tends to 0 uniformly in x≥0.Proof.
We change a bit the standard proof of local limit theorem. LetpN(z) be the density of the standard deviation(47)SN,λN-NmλNNσλN,where SN,λ=ξ1,λ+⋯+ξN,λ is the sum independent random variables having density (25). Let q(z) be the standard Gaussian density. The inverse Fourier transform gives(48)supz∈R⁡pNz-qz≤12π∫-∞∞ψλNNtNσλN-12πe-t2/2dt,where ψλ(t)=ϕλ(t)e-itmλ. Denote(49)I1=12π∫t≤NaλN-1ψλNNtNσλN-12πe-t2/2dt,I2=12π∫t>NaλN-1ψλNtNσλNNdt,I3=12π3/2∫t>NaλN-1e-t2/2dt,where aλ=E|ξλ-mλ|4/σλ4 is bounded uniformly in λ (Lemma 5, part (1)). Then(50)supz∈R⁡pNz-qz≤I1+I2+I3.For I1 we have the estimate (see [21, page 109, lemma  1])(51)I1≤aλN2πN∫t≤NaλN-1t3e-t3/3dt≤CaλNN.For I2 we have by parts (2) and (3) of Lemma 5,(52)I2=12π∫t>NaλN-1ϕλNtNσλNNdt≤N-1/2γN-q∫-∞∞ϕλNtqdt≤CN-1/2γN-qNqαfor N being sufficiently large, where γ<1 and q>1.
The estimate forI3 is trivial. Thus,(53)pNx=qx+ON-1/2,where O(N-1/2) does not depend on x∈R. Lemma follows as(54)hλN⋆Nx=1NσλNpNx-1NσλN.The lemma is proved.Lemma 7.
Assume conditions (1)–(3) of Lemma5. Then the conditional variance(55)dN=DN+oσλN2,where(56)DN=∫01x-1N2hλNxexp⁡-x-1/N22σλN2N-1dx.Proof.
By Lemma6 we have(57)hλN⋆N-11-x=1σλN2πN-1exp⁡-x-1/N22σλN2N-1+o1,(58)hλN⋆N1=1σλN2πN1+o1,where mλN=1/N. The division gives(59)hλN⋆N-11-xhλN⋆N1=NN-1exp⁡-x-1/N22σλN2N-1+o1,where the term o(1) tends to 0 uniformly in x∈[0,1]. By (27) the conditional variance is(60)dN=∫01x-1N2hλNxhλN⋆N-11-xhλN⋆N1dx.Substituting (59) into this expression we get(61)dN=∫01x-1N2hλNxexp⁡-x-1/N22σλN2N-1dx+oσλN2.The lemma is proved.By Lemma4σλN2~α-1N-2 as N→∞. Thus(62)dN=DN+oN-2,where(63)DN=∫01x-1N2hλNxexp⁡-x-1/N22σλN2N-1dx.Lemma 8.
AsN→∞,(64)DN~c1N-2.Proof.
By definition (25) and Lemma 4 as N→∞,(65)DN~αNαc0Γα∫01x-1N2e-αNxgxexp⁡-αNx-1/N22dx.Because of g(x)~c0xα-1 as x→0 we have(66)DN~αNαe-αΓα∫0N-1x-1N2xα-1dx.Changing variable y=Nx we have(67)αNαe-αΓα∫0N-1x-1N2xα-1dx=ααe-αΓαN2∫011-y2yα-1dywhich gives the lemma and the theorem.


#### 3.2.2. Coulomb Case

Again we introduce the exponential family of densities(68)hλx=e-λxgxZ-1λ,λ≥0,where the function g(x) satisfies condition (22) with β>0 and(69)Zλ=∫01e-λxgxdx.We will use modified Bessel function of the second kind defined as follows:(70)Kαz=12∫0∞xα-1e-z/2x+1/xdx,where α∈R and z>0. For Kα(z) we know the asymptotic expansion as z→∞:(71)Kαz~π2e-zz1+4α2-18z+⋯.Let againξλ be a random variable with the density hλ(x) and put mλ=Eξλ and σλ2=Dξλ.Lemma 9.
(1) Asλ→∞,(72)Zλ~c0π2βλα/2e-2λβλβ1/4.
(2) Asλ→∞,(73)mλ~βλ,σλ2~12β1/2λ3/2.
(3) There exists a uniqueλN such that mλN=1/N such that λN~βN2 and σλN2~2-1β-2N-3 as N→∞.Proof.
(1) We can write(74)Zλ=I1λ+I2λ,where(75)I1λ=∫0λ-ϵe-λxgxdx,I2λ=∫λ-ϵ1e-λxgxdxand ϵ>0 is small enough. By (22)(76)I1λ~c0I1′λ=c0∫0λ-ϵe-λx-β/xxα-1dxas λ→∞. One can write(77)I1′λ=Z1λ-Z1′λ,where(78)Z1λ=∫0∞e-λx-β/xxα-1dx,Z1′λ=∫λ-ϵ∞e-λx-β/xxα-1dx.Find asymptotics of Z1(λ) as λ→∞. Changing variable y=β-1x gives(79)Z1λ=βα∫0∞e-λβy-1/yyα-1dyand changing variable t=(2/z)x in (70) gives(80)Kαz=2-α-1zα∫0∞tα-1e-z2/4t-1/tdt.
Putz=2λβ. Then(81)Z1λ=2βα/2Kα2λβλα/2and using (71) we get(82)Z1λ~π2βλα/2e-2λβλβ1/4,λ⟶∞.Taking into account I2(λ)=O(e-λ1-ϵ) and Z1′(λ)=O(e-λ1-ϵ) for some small enough ϵ>0 we come to (72).
(2) Mathematical expectation is(83)mλ=Z-1λ∫01xe-λxgxdx.By part (1) of this lemma we have(84)∫01xe-λxgxdx~π2βλα+1/2e-2λβλβ1/4,λ⟶∞.So(85)mλ~βλ,λ⟶∞.
Covariance is equal to(86)σλ2=Z-1λ∫01x2e-λxgxdx-Z-2λ∫01xe-λxgxdx2.It follows from part (1) that, as λ→∞,(87)σλ2~βλKα+22λβKα2λβ-Kα+12λβKα2λβ2.Using asymptotic expansion (71) we get(88)σλ2~12β1/2λ3/2as λ→∞.
(3) It follows from continuity ofmλ as function of λ and (85) that there exists λN such that mλN=1/N and then λN~βN2 as N→∞. By (88)(89)σλN2~12β2N3,N⟶∞.Lemma is proved.Lemma 10.
The exponential family (68) has the following properties: (1)
The normalized momentaλ=E|ξλ-mλ|4/σλ4 is bounded uniformly in λ>0.(2)
For anyδ>0 there exists λ0>0 such that supλ>λ0supt>δϕλ(t/σλ)<1.(3)
For someq≥1, as λ→∞,(90)∫-∞∞ϕλtqdt=Oλqα.Proof.
It is similar to Lemma5.
Using Lemmas6, 7, and 9 we find(91)dN=DN+oN-3,where(92)DN=∫01x-1N2hλNxexp⁡-x-1/N22σλN2N-1dx.By Lemma 9λN~βN2 and σλN2~2-1β-2N-3 as N→∞, then(93)DN~Z-1λN∫01x-1N2e-βN2xgxxα-1exp⁡-β2N2x-1N2dx.We split the integral in (93) into two integrals: (94)DN1=∫0N-1/2x-1N2e-βN2xgxxα-1exp⁡-β2N2x-1N2dx,DN2=∫N-1/21x-1N2e-βN2xgxxα-1exp⁡-β2N2x-1N2dx.By condition (22)(95)Z-1λNDN1~Z-1λNc0∫0N-1/2x-1N2e-β1/x+N2xxα-1exp⁡-β2N2x-1N2dx.To find the asymptotics of DN(1) we use Laplace’s method. Consider the function s(x)=β1/x+N2x. Its derivative(96)s′x=β-1x2+N2equals 0 at the point N-1. The second derivative(97)s′′x=2βx3.The function s(x) can be expanded at the neighborhood of N-1 using Taylor’s formula:(98)sx=2βN+βN3x-1N2+Ox-1N3.By (72) we have(99)Z-1λN~c0-12βπe2βNNα+1/2and so we get, as N→∞,(100)Z-1λNDN1~c0-12βπe2βNNα+1/2c0∫01x-1N2N-α+1e-2βN-βN3x-1/N2dx.After cancellations(101)Z-1λNDN1~2βπN3/2∫01x-1N2e-N3x-1/N2dx~β2N-3as N→∞. For the second integral DN(2) we have, as N→∞,(102)Z-1λNDN2=Oe-βN.So(103)DN~β2N-3,N⟶∞.Theorem is proved.


## 3.1. Results

We consider the densities having the following asymptotic behaviour asx→0:(22)gx~c0xα-1e-β/x,α∈R,β≥0.Theorem 2.
Under this condition and ifα>0 and β=0, as N→∞,(23)dN~c1N-2for some constant c1=c1(α)>0 depending only on α.Theorem 3.
Ifβ>0, then for any α∈R, as N→∞,(24)dN~c1N-3for some constant c1=c1(β)>0 depending only on β.It is of interest to know the behaviour of the covariance for densities with hyperexponential decrease at zero.


## 3.2. Proofs

### 3.2.1. General Power Asymptotics

We will prove here Theorem2. Instead of one density g(x) it is useful to consider the family of densities (such trick has been used in some large deviations problems, see [18, 19])(25)hλx=e-λxgxz-1λ,where λ≥0 and(26)zλ=∫01e-λxgxdx.Let ξλ,k be random variables with density hλ(x). Put mλ=Eξλ,k, σλ2=Dξλ,k and denote the conditional densities of ξλ,k under the condition that SN=1:(27)fλx=hλxhλ⋆N-11-xhλ⋆N1=gxg⋆n-11-xg⋆n1,x∈0,1.It is easy to check that fλ(x) in fact does not depend on λ.Lemma 4.
(1)mλ~αλ-1 and σλ2~αλ-2 as λ→∞. (2) there exists a unique λN, such that mλN=1/N. Then λN~αN, σλn2~α-1N-2 as N→∞.Proof.
(1) By abelian theorem, see [20, page 445, Theorem  3], and the condition g(x)~c0xα-1 as x→0 we have as λ→∞(28)zλ~Γαc0λN,-z′λ~Γα+1c0λN+1,z′′λ~Γα+2c0λN+2.It follows that(29)mλ=-z′λzλ~Γα+1Γαλ=αλ,σλ2=c′′λcλ-c′λcλ2~Γα+2Γα-Γα+1Γα2λ-2=α2λ2as λ→∞.
(2) The functionmλ is monotonically decreasing in λ. Thus for any N there exists λN such that(30)mλN=1N.From (1) it follows that λN~αN and σλN2~α-1N-2 as N→∞.Letϕλ(t) be the characteristic function of ξλ.Lemma 5.
The family of densities (25) has the following properties: (1)
The normalized momentaλ=E|ξλ-mλ|4/σλ4 is bounded uniformly in λ>0.(2)
For anyδ>0 there exists λ0>0 such that supλ>λ0supt>δϕλ(t/σλ)<1.(3)
For someq≥1(31)∫-∞∞ϕλtqdt=Oλqα.Proof.
(1) It is similar to the proof of (1) in Lemma4.
(2) Put(32)ft,λ=ϕλtσλ-1=z-1λ∫01eitσλ-1xe-λxgxdx.Let us show that for some δ>0(33)ft,λ≤11-itα-1/2α+Oe-δλas λ→∞. For this we can write f(t,λ) as(34)ft,λ=J1λ+J2λ,where(35)J1λ=z-1λ∫0λ-1/2eitσλ-1xe-λxgxdx,J2λ=z-1λ∫λ-1/21eitσλ-1xe-λxgxdx.Taking into account that(36)zλ~c0Γαλ-α,σλ2~αλ-2,λ⟶∞and condition g(x)~c0xα-1 as x→0 we have that as λ→∞(37)J1λ~Γα-1λα∫0λ-1/2eitα-1/2λxe-λxxα-1dx.Putting y=λx in the last integral we get(38)Γα-1λα∫0λ-1/2eitα-1/2λxe-λxxα-1dx=Γα-1∫0λ1/2eitα-1/2ye-yyα-1dy.As (39)Γα-1∫0λ1/2eitα-1/2ye-yyα-1dy=Γα-1∫0∞eitα-1/2ye-yyα-1dy-Γα-1∫λ1/2∞eitα-1/2ye-yyα-1dy,Γα-1∫0∞eitα-1/2ye-yyα-1dy=11-itα-1/2α,Γα-1∫λ1/2∞eitα-1/2ye-yyα-1dy=Oe-δλfor some δ>0, then(40)J1λ≤11-itα-1/2α+Oe-δλ.For J2(λ) we get the estimate(41)J2λ=c-1λ∫λ-1/21eitσλ-1xe-λxgxdx≤Cλαe-λ=Oe-δλ.From these estimates (33) follows.
(3) Note that always(42)∫01hλxpdx<∞for some p>1. Without loss of generality one can assume that 1<p≤2. By Hausdorff-Young inequality(43)∫-∞∞ϕλtqdt1/q≤12π∫01hλxpdx1/p,where 1<p≤2 and 1/p+1/q=1. As(44)12π∫01hλxpdt1/p=z-1λ12π∫01e-λxgxpdt1/p≤z-1λ12π∫01gxpdt1/pand z(λ)~Cλ-α, then(45)∫-∞∞ϕλtqdt=Oλqα.Lemma 6.
Assume conditions (1)–(3) of Lemma5 and that λN are defined by the condition mλN=1/N. Then(46)hλN⋆Nx=1σλN2πNexp⁡-x-122σλN2N+o1,where o(1) tends to 0 uniformly in x≥0.Proof.
We change a bit the standard proof of local limit theorem. LetpN(z) be the density of the standard deviation(47)SN,λN-NmλNNσλN,where SN,λ=ξ1,λ+⋯+ξN,λ is the sum independent random variables having density (25). Let q(z) be the standard Gaussian density. The inverse Fourier transform gives(48)supz∈R⁡pNz-qz≤12π∫-∞∞ψλNNtNσλN-12πe-t2/2dt,where ψλ(t)=ϕλ(t)e-itmλ. Denote(49)I1=12π∫t≤NaλN-1ψλNNtNσλN-12πe-t2/2dt,I2=12π∫t>NaλN-1ψλNtNσλNNdt,I3=12π3/2∫t>NaλN-1e-t2/2dt,where aλ=E|ξλ-mλ|4/σλ4 is bounded uniformly in λ (Lemma 5, part (1)). Then(50)supz∈R⁡pNz-qz≤I1+I2+I3.For I1 we have the estimate (see [21, page 109, lemma  1])(51)I1≤aλN2πN∫t≤NaλN-1t3e-t3/3dt≤CaλNN.For I2 we have by parts (2) and (3) of Lemma 5,(52)I2=12π∫t>NaλN-1ϕλNtNσλNNdt≤N-1/2γN-q∫-∞∞ϕλNtqdt≤CN-1/2γN-qNqαfor N being sufficiently large, where γ<1 and q>1.
The estimate forI3 is trivial. Thus,(53)pNx=qx+ON-1/2,where O(N-1/2) does not depend on x∈R. Lemma follows as(54)hλN⋆Nx=1NσλNpNx-1NσλN.The lemma is proved.Lemma 7.
Assume conditions (1)–(3) of Lemma5. Then the conditional variance(55)dN=DN+oσλN2,where(56)DN=∫01x-1N2hλNxexp⁡-x-1/N22σλN2N-1dx.Proof.
By Lemma6 we have(57)hλN⋆N-11-x=1σλN2πN-1exp⁡-x-1/N22σλN2N-1+o1,(58)hλN⋆N1=1σλN2πN1+o1,where mλN=1/N. The division gives(59)hλN⋆N-11-xhλN⋆N1=NN-1exp⁡-x-1/N22σλN2N-1+o1,where the term o(1) tends to 0 uniformly in x∈[0,1]. By (27) the conditional variance is(60)dN=∫01x-1N2hλNxhλN⋆N-11-xhλN⋆N1dx.Substituting (59) into this expression we get(61)dN=∫01x-1N2hλNxexp⁡-x-1/N22σλN2N-1dx+oσλN2.The lemma is proved.By Lemma4σλN2~α-1N-2 as N→∞. Thus(62)dN=DN+oN-2,where(63)DN=∫01x-1N2hλNxexp⁡-x-1/N22σλN2N-1dx.Lemma 8.
AsN→∞,(64)DN~c1N-2.Proof.
By definition (25) and Lemma 4 as N→∞,(65)DN~αNαc0Γα∫01x-1N2e-αNxgxexp⁡-αNx-1/N22dx.Because of g(x)~c0xα-1 as x→0 we have(66)DN~αNαe-αΓα∫0N-1x-1N2xα-1dx.Changing variable y=Nx we have(67)αNαe-αΓα∫0N-1x-1N2xα-1dx=ααe-αΓαN2∫011-y2yα-1dywhich gives the lemma and the theorem.


### 3.2.2. Coulomb Case

Again we introduce the exponential family of densities(68)hλx=e-λxgxZ-1λ,λ≥0,where the function g(x) satisfies condition (22) with β>0 and(69)Zλ=∫01e-λxgxdx.We will use modified Bessel function of the second kind defined as follows:(70)Kαz=12∫0∞xα-1e-z/2x+1/xdx,where α∈R and z>0. For Kα(z) we know the asymptotic expansion as z→∞:(71)Kαz~π2e-zz1+4α2-18z+⋯.Let againξλ be a random variable with the density hλ(x) and put mλ=Eξλ and σλ2=Dξλ.Lemma 9.
(1) Asλ→∞,(72)Zλ~c0π2βλα/2e-2λβλβ1/4.
(2) Asλ→∞,(73)mλ~βλ,σλ2~12β1/2λ3/2.
(3) There exists a uniqueλN such that mλN=1/N such that λN~βN2 and σλN2~2-1β-2N-3 as N→∞.Proof.
(1) We can write(74)Zλ=I1λ+I2λ,where(75)I1λ=∫0λ-ϵe-λxgxdx,I2λ=∫λ-ϵ1e-λxgxdxand ϵ>0 is small enough. By (22)(76)I1λ~c0I1′λ=c0∫0λ-ϵe-λx-β/xxα-1dxas λ→∞. One can write(77)I1′λ=Z1λ-Z1′λ,where(78)Z1λ=∫0∞e-λx-β/xxα-1dx,Z1′λ=∫λ-ϵ∞e-λx-β/xxα-1dx.Find asymptotics of Z1(λ) as λ→∞. Changing variable y=β-1x gives(79)Z1λ=βα∫0∞e-λβy-1/yyα-1dyand changing variable t=(2/z)x in (70) gives(80)Kαz=2-α-1zα∫0∞tα-1e-z2/4t-1/tdt.
Putz=2λβ. Then(81)Z1λ=2βα/2Kα2λβλα/2and using (71) we get(82)Z1λ~π2βλα/2e-2λβλβ1/4,λ⟶∞.Taking into account I2(λ)=O(e-λ1-ϵ) and Z1′(λ)=O(e-λ1-ϵ) for some small enough ϵ>0 we come to (72).
(2) Mathematical expectation is(83)mλ=Z-1λ∫01xe-λxgxdx.By part (1) of this lemma we have(84)∫01xe-λxgxdx~π2βλα+1/2e-2λβλβ1/4,λ⟶∞.So(85)mλ~βλ,λ⟶∞.
Covariance is equal to(86)σλ2=Z-1λ∫01x2e-λxgxdx-Z-2λ∫01xe-λxgxdx2.It follows from part (1) that, as λ→∞,(87)σλ2~βλKα+22λβKα2λβ-Kα+12λβKα2λβ2.Using asymptotic expansion (71) we get(88)σλ2~12β1/2λ3/2as λ→∞.
(3) It follows from continuity ofmλ as function of λ and (85) that there exists λN such that mλN=1/N and then λN~βN2 as N→∞. By (88)(89)σλN2~12β2N3,N⟶∞.Lemma is proved.Lemma 10.
The exponential family (68) has the following properties: (1)
The normalized momentaλ=E|ξλ-mλ|4/σλ4 is bounded uniformly in λ>0.(2)
For anyδ>0 there exists λ0>0 such that supλ>λ0supt>δϕλ(t/σλ)<1.(3)
For someq≥1, as λ→∞,(90)∫-∞∞ϕλtqdt=Oλqα.Proof.
It is similar to Lemma5.
Using Lemmas6, 7, and 9 we find(91)dN=DN+oN-3,where(92)DN=∫01x-1N2hλNxexp⁡-x-1/N22σλN2N-1dx.By Lemma 9λN~βN2 and σλN2~2-1β-2N-3 as N→∞, then(93)DN~Z-1λN∫01x-1N2e-βN2xgxxα-1exp⁡-β2N2x-1N2dx.We split the integral in (93) into two integrals: (94)DN1=∫0N-1/2x-1N2e-βN2xgxxα-1exp⁡-β2N2x-1N2dx,DN2=∫N-1/21x-1N2e-βN2xgxxα-1exp⁡-β2N2x-1N2dx.By condition (22)(95)Z-1λNDN1~Z-1λNc0∫0N-1/2x-1N2e-β1/x+N2xxα-1exp⁡-β2N2x-1N2dx.To find the asymptotics of DN(1) we use Laplace’s method. Consider the function s(x)=β1/x+N2x. Its derivative(96)s′x=β-1x2+N2equals 0 at the point N-1. The second derivative(97)s′′x=2βx3.The function s(x) can be expanded at the neighborhood of N-1 using Taylor’s formula:(98)sx=2βN+βN3x-1N2+Ox-1N3.By (72) we have(99)Z-1λN~c0-12βπe2βNNα+1/2and so we get, as N→∞,(100)Z-1λNDN1~c0-12βπe2βNNα+1/2c0∫01x-1N2N-α+1e-2βN-βN3x-1/N2dx.After cancellations(101)Z-1λNDN1~2βπN3/2∫01x-1N2e-N3x-1/N2dx~β2N-3as N→∞. For the second integral DN(2) we have, as N→∞,(102)Z-1λNDN2=Oe-βN.So(103)DN~β2N-3,N⟶∞.Theorem is proved.


## 3.2.1. General Power Asymptotics

We will prove here Theorem2. Instead of one density g(x) it is useful to consider the family of densities (such trick has been used in some large deviations problems, see [18, 19])(25)hλx=e-λxgxz-1λ,where λ≥0 and(26)zλ=∫01e-λxgxdx.Let ξλ,k be random variables with density hλ(x). Put mλ=Eξλ,k, σλ2=Dξλ,k and denote the conditional densities of ξλ,k under the condition that SN=1:(27)fλx=hλxhλ⋆N-11-xhλ⋆N1=gxg⋆n-11-xg⋆n1,x∈0,1.It is easy to check that fλ(x) in fact does not depend on λ.Lemma 4.
(1)mλ~αλ-1 and σλ2~αλ-2 as λ→∞. (2) there exists a unique λN, such that mλN=1/N. Then λN~αN, σλn2~α-1N-2 as N→∞.Proof.
(1) By abelian theorem, see [20, page 445, Theorem  3], and the condition g(x)~c0xα-1 as x→0 we have as λ→∞(28)zλ~Γαc0λN,-z′λ~Γα+1c0λN+1,z′′λ~Γα+2c0λN+2.It follows that(29)mλ=-z′λzλ~Γα+1Γαλ=αλ,σλ2=c′′λcλ-c′λcλ2~Γα+2Γα-Γα+1Γα2λ-2=α2λ2as λ→∞.
(2) The functionmλ is monotonically decreasing in λ. Thus for any N there exists λN such that(30)mλN=1N.From (1) it follows that λN~αN and σλN2~α-1N-2 as N→∞.Letϕλ(t) be the characteristic function of ξλ.Lemma 5.
The family of densities (25) has the following properties: (1)
The normalized momentaλ=E|ξλ-mλ|4/σλ4 is bounded uniformly in λ>0.(2)
For anyδ>0 there exists λ0>0 such that supλ>λ0supt>δϕλ(t/σλ)<1.(3)
For someq≥1(31)∫-∞∞ϕλtqdt=Oλqα.Proof.
(1) It is similar to the proof of (1) in Lemma4.
(2) Put(32)ft,λ=ϕλtσλ-1=z-1λ∫01eitσλ-1xe-λxgxdx.Let us show that for some δ>0(33)ft,λ≤11-itα-1/2α+Oe-δλas λ→∞. For this we can write f(t,λ) as(34)ft,λ=J1λ+J2λ,where(35)J1λ=z-1λ∫0λ-1/2eitσλ-1xe-λxgxdx,J2λ=z-1λ∫λ-1/21eitσλ-1xe-λxgxdx.Taking into account that(36)zλ~c0Γαλ-α,σλ2~αλ-2,λ⟶∞and condition g(x)~c0xα-1 as x→0 we have that as λ→∞(37)J1λ~Γα-1λα∫0λ-1/2eitα-1/2λxe-λxxα-1dx.Putting y=λx in the last integral we get(38)Γα-1λα∫0λ-1/2eitα-1/2λxe-λxxα-1dx=Γα-1∫0λ1/2eitα-1/2ye-yyα-1dy.As (39)Γα-1∫0λ1/2eitα-1/2ye-yyα-1dy=Γα-1∫0∞eitα-1/2ye-yyα-1dy-Γα-1∫λ1/2∞eitα-1/2ye-yyα-1dy,Γα-1∫0∞eitα-1/2ye-yyα-1dy=11-itα-1/2α,Γα-1∫λ1/2∞eitα-1/2ye-yyα-1dy=Oe-δλfor some δ>0, then(40)J1λ≤11-itα-1/2α+Oe-δλ.For J2(λ) we get the estimate(41)J2λ=c-1λ∫λ-1/21eitσλ-1xe-λxgxdx≤Cλαe-λ=Oe-δλ.From these estimates (33) follows.
(3) Note that always(42)∫01hλxpdx<∞for some p>1. Without loss of generality one can assume that 1<p≤2. By Hausdorff-Young inequality(43)∫-∞∞ϕλtqdt1/q≤12π∫01hλxpdx1/p,where 1<p≤2 and 1/p+1/q=1. As(44)12π∫01hλxpdt1/p=z-1λ12π∫01e-λxgxpdt1/p≤z-1λ12π∫01gxpdt1/pand z(λ)~Cλ-α, then(45)∫-∞∞ϕλtqdt=Oλqα.Lemma 6.
Assume conditions (1)–(3) of Lemma5 and that λN are defined by the condition mλN=1/N. Then(46)hλN⋆Nx=1σλN2πNexp⁡-x-122σλN2N+o1,where o(1) tends to 0 uniformly in x≥0.Proof.
We change a bit the standard proof of local limit theorem. LetpN(z) be the density of the standard deviation(47)SN,λN-NmλNNσλN,where SN,λ=ξ1,λ+⋯+ξN,λ is the sum independent random variables having density (25). Let q(z) be the standard Gaussian density. The inverse Fourier transform gives(48)supz∈R⁡pNz-qz≤12π∫-∞∞ψλNNtNσλN-12πe-t2/2dt,where ψλ(t)=ϕλ(t)e-itmλ. Denote(49)I1=12π∫t≤NaλN-1ψλNNtNσλN-12πe-t2/2dt,I2=12π∫t>NaλN-1ψλNtNσλNNdt,I3=12π3/2∫t>NaλN-1e-t2/2dt,where aλ=E|ξλ-mλ|4/σλ4 is bounded uniformly in λ (Lemma 5, part (1)). Then(50)supz∈R⁡pNz-qz≤I1+I2+I3.For I1 we have the estimate (see [21, page 109, lemma  1])(51)I1≤aλN2πN∫t≤NaλN-1t3e-t3/3dt≤CaλNN.For I2 we have by parts (2) and (3) of Lemma 5,(52)I2=12π∫t>NaλN-1ϕλNtNσλNNdt≤N-1/2γN-q∫-∞∞ϕλNtqdt≤CN-1/2γN-qNqαfor N being sufficiently large, where γ<1 and q>1.
The estimate forI3 is trivial. Thus,(53)pNx=qx+ON-1/2,where O(N-1/2) does not depend on x∈R. Lemma follows as(54)hλN⋆Nx=1NσλNpNx-1NσλN.The lemma is proved.Lemma 7.
Assume conditions (1)–(3) of Lemma5. Then the conditional variance(55)dN=DN+oσλN2,where(56)DN=∫01x-1N2hλNxexp⁡-x-1/N22σλN2N-1dx.Proof.
By Lemma6 we have(57)hλN⋆N-11-x=1σλN2πN-1exp⁡-x-1/N22σλN2N-1+o1,(58)hλN⋆N1=1σλN2πN1+o1,where mλN=1/N. The division gives(59)hλN⋆N-11-xhλN⋆N1=NN-1exp⁡-x-1/N22σλN2N-1+o1,where the term o(1) tends to 0 uniformly in x∈[0,1]. By (27) the conditional variance is(60)dN=∫01x-1N2hλNxhλN⋆N-11-xhλN⋆N1dx.Substituting (59) into this expression we get(61)dN=∫01x-1N2hλNxexp⁡-x-1/N22σλN2N-1dx+oσλN2.The lemma is proved.By Lemma4σλN2~α-1N-2 as N→∞. Thus(62)dN=DN+oN-2,where(63)DN=∫01x-1N2hλNxexp⁡-x-1/N22σλN2N-1dx.Lemma 8.
AsN→∞,(64)DN~c1N-2.Proof.
By definition (25) and Lemma 4 as N→∞,(65)DN~αNαc0Γα∫01x-1N2e-αNxgxexp⁡-αNx-1/N22dx.Because of g(x)~c0xα-1 as x→0 we have(66)DN~αNαe-αΓα∫0N-1x-1N2xα-1dx.Changing variable y=Nx we have(67)αNαe-αΓα∫0N-1x-1N2xα-1dx=ααe-αΓαN2∫011-y2yα-1dywhich gives the lemma and the theorem.


## 3.2.2. Coulomb Case

Again we introduce the exponential family of densities(68)hλx=e-λxgxZ-1λ,λ≥0,where the function g(x) satisfies condition (22) with β>0 and(69)Zλ=∫01e-λxgxdx.We will use modified Bessel function of the second kind defined as follows:(70)Kαz=12∫0∞xα-1e-z/2x+1/xdx,where α∈R and z>0. For Kα(z) we know the asymptotic expansion as z→∞:(71)Kαz~π2e-zz1+4α2-18z+⋯.Let againξλ be a random variable with the density hλ(x) and put mλ=Eξλ and σλ2=Dξλ.Lemma 9.
(1) Asλ→∞,(72)Zλ~c0π2βλα/2e-2λβλβ1/4.
(2) Asλ→∞,(73)mλ~βλ,σλ2~12β1/2λ3/2.
(3) There exists a uniqueλN such that mλN=1/N such that λN~βN2 and σλN2~2-1β-2N-3 as N→∞.Proof.
(1) We can write(74)Zλ=I1λ+I2λ,where(75)I1λ=∫0λ-ϵe-λxgxdx,I2λ=∫λ-ϵ1e-λxgxdxand ϵ>0 is small enough. By (22)(76)I1λ~c0I1′λ=c0∫0λ-ϵe-λx-β/xxα-1dxas λ→∞. One can write(77)I1′λ=Z1λ-Z1′λ,where(78)Z1λ=∫0∞e-λx-β/xxα-1dx,Z1′λ=∫λ-ϵ∞e-λx-β/xxα-1dx.Find asymptotics of Z1(λ) as λ→∞. Changing variable y=β-1x gives(79)Z1λ=βα∫0∞e-λβy-1/yyα-1dyand changing variable t=(2/z)x in (70) gives(80)Kαz=2-α-1zα∫0∞tα-1e-z2/4t-1/tdt.
Putz=2λβ. Then(81)Z1λ=2βα/2Kα2λβλα/2and using (71) we get(82)Z1λ~π2βλα/2e-2λβλβ1/4,λ⟶∞.Taking into account I2(λ)=O(e-λ1-ϵ) and Z1′(λ)=O(e-λ1-ϵ) for some small enough ϵ>0 we come to (72).
(2) Mathematical expectation is(83)mλ=Z-1λ∫01xe-λxgxdx.By part (1) of this lemma we have(84)∫01xe-λxgxdx~π2βλα+1/2e-2λβλβ1/4,λ⟶∞.So(85)mλ~βλ,λ⟶∞.
Covariance is equal to(86)σλ2=Z-1λ∫01x2e-λxgxdx-Z-2λ∫01xe-λxgxdx2.It follows from part (1) that, as λ→∞,(87)σλ2~βλKα+22λβKα2λβ-Kα+12λβKα2λβ2.Using asymptotic expansion (71) we get(88)σλ2~12β1/2λ3/2as λ→∞.
(3) It follows from continuity ofmλ as function of λ and (85) that there exists λN such that mλN=1/N and then λN~βN2 as N→∞. By (88)(89)σλN2~12β2N3,N⟶∞.Lemma is proved.Lemma 10.
The exponential family (68) has the following properties: (1)
The normalized momentaλ=E|ξλ-mλ|4/σλ4 is bounded uniformly in λ>0.(2)
For anyδ>0 there exists λ0>0 such that supλ>λ0supt>δϕλ(t/σλ)<1.(3)
For someq≥1, as λ→∞,(90)∫-∞∞ϕλtqdt=Oλqα.Proof.
It is similar to Lemma5.
Using Lemmas6, 7, and 9 we find(91)dN=DN+oN-3,where(92)DN=∫01x-1N2hλNxexp⁡-x-1/N22σλN2N-1dx.By Lemma 9λN~βN2 and σλN2~2-1β-2N-3 as N→∞, then(93)DN~Z-1λN∫01x-1N2e-βN2xgxxα-1exp⁡-β2N2x-1N2dx.We split the integral in (93) into two integrals: (94)DN1=∫0N-1/2x-1N2e-βN2xgxxα-1exp⁡-β2N2x-1N2dx,DN2=∫N-1/21x-1N2e-βN2xgxxα-1exp⁡-β2N2x-1N2dx.By condition (22)(95)Z-1λNDN1~Z-1λNc0∫0N-1/2x-1N2e-β1/x+N2xxα-1exp⁡-β2N2x-1N2dx.To find the asymptotics of DN(1) we use Laplace’s method. Consider the function s(x)=β1/x+N2x. Its derivative(96)s′x=β-1x2+N2equals 0 at the point N-1. The second derivative(97)s′′x=2βx3.The function s(x) can be expanded at the neighborhood of N-1 using Taylor’s formula:(98)sx=2βN+βN3x-1N2+Ox-1N3.By (72) we have(99)Z-1λN~c0-12βπe2βNNα+1/2and so we get, as N→∞,(100)Z-1λNDN1~c0-12βπe2βNNα+1/2c0∫01x-1N2N-α+1e-2βN-βN3x-1/N2dx.After cancellations(101)Z-1λNDN1~2βπN3/2∫01x-1N2e-N3x-1/N2dx~β2N-3as N→∞. For the second integral DN(2) we have, as N→∞,(102)Z-1λNDN2=Oe-βN.So(103)DN~β2N-3,N⟶∞.Theorem is proved.



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