H M Frazão

José Geraldo Peixoto De Faria

G Q Pellegrino

M C Nemes

Quantum phase transition in an

come   L'archive ouverte pluridisciplinaire

I. INTRODUCTION

One issue of great interest in condensed matter physics today is quantum phase transition (QPT) [START_REF] Sachdev | Quantum Phase Transitions[END_REF]. Differently from the usual thermodynamic phase transition, which is guided by thermodynamic fluctuations and characterized by a critical temperature, a QPT can be observed for T = 0 and is conducted by quantum fluctuations. The change in the system due to a QPT is observed when some parameter of the Hamiltonian is varied, rather than temperature as in the thermodynamic phase transition. As a simple example, we have the quantum Ising model in a transverse field, where a QPT is observed between ferromagnetic and paramagnetic phases when the intensity of the applied field is varied [START_REF] Sachdev | Quantum Phase Transitions[END_REF][START_REF] Zurek | [END_REF]. As this example illustrates, the study of QPT conveys a better understanding of the complex behavior shown by many-body systems. This study can cross very different systems as, for example, systems involving light-matter interactions, such as cavity arrays coupled by optical fibers [3], or a two-species condensate of interacting bosons trapped in optical lattices [4]. The next paragraphs present other instances where QPT in many-body problems is considered and which are of interest for this work.

One of the systems under intense attention in recent decades is the semiconductor microcavity [START_REF] Kavokin | Microcavities[END_REF]. In this system, the interaction between cavity photons and excitons belonging to the semiconductor gives rise to a new quasi-particle called exciton-polariton [START_REF] Kavokin | Thin Films and Nanostructures: Cavity Polaritons[END_REF]. There are many interesting features shown by this system as, for example, the superfluidity [START_REF] Keeling | [END_REF],

and the generation of a Bose-Einstein condensate (BEC) in a solid state system [8] (which occurs even at room temperature [9]). Another interesting feature is the so-called magic angle configuration in which we observe a parametric amplification of the emitted light [10].

One possible theoretical description of this feature is given by considering only three modes for the exciton-polaritons, denominated pump, signal and idler [11]. The evidences pointed by experiment [10] suggest that only these three modes are coherently and macroscopicaly populated [START_REF] Kavokin | Microcavities[END_REF]11]. In this situation one can approximate the Hamiltonian of the system by an effective three-mode Hamiltonian as we do in this work.

The experimental observation of QPT in the scope of condensed matter physics has been boosted by the development of techniques that allowed for the storage and handling of matter at the atomic level. In the past two decades, intense research has been done on systems of interacting trapped bosons, especially cold atoms trapped in optical lattices [12].

The interest in such systems is assigned to the their capacity to simulate many phenomena predicted to occur in arrangements of trapped atoms, such as the BEC itself [13], solitons [14], bosonic Josephson effect, and nonlinear oscillations [15], and also a QPT from a Mottinsulator-like phase to a superfluid-like one [START_REF] Sachdev | Quantum Phase Transitions[END_REF]16]. For a two-well condensate [17] or a three-well condensate [18,19], the transition is between two dynamical regimes: macroscopic self-trapping and Rabi (or Josephson) oscillations [20]. These kinds of systems are generally described by a Bose-Hubbard-type Hamiltonian [21][22][23]. In this work we study an effective three-mode Hamiltonian essentially different from a Bose-Hubbard Hamiltonian but which exhibits a QPT between two phases: the macroscopic self-trapping (MST), characterized by the vanishing of the tunneling between the modes, and a regime of oscillations (RO), where tunneling is present.

In recent years, also classical analyses involving the thermodynamic limit of some quantum many-body models have contributed to the investigations of QPT. These analyses come to add themselves to other valuable tools to investigate QPT, many of them referring to the properties of the spectrum of the corresponding quantum Hamiltonian, as the level approximation (crossing), or to the measurement of entanglement of the ground state [24] near the critical point. Different systems were studied in this quantum-classical context, e.g., the Lipkin model [25] and the pairing model [26] (both in nuclear physics), the Dicke model for superradiance in quantum optics [27], excitons in semiconductor bilayer electron systems [28], ultracold Bose gases trapped in multiple wells [19], among others [29]. In parallel with the quantum treatment, we perform as well the classical limit analysis to study QPT in the effective three-mode model for a system of interacting bosons. Our analysis results in a close analogy between some aspects of the spectrum of the quantum Hamiltonian and classical properties of the corresponding thermodynamic limit of the model. As an example, the different phases in the quantum regime are related to the existence of closed orbits in the classical thermodynamic limit. In fact, the MST phase corresponds to the situation in which there are no closed orbits in phase space, whereas the RO phase is characterized by the presence of such orbits. In this way, separatrices delimiting regions of the phase space characterized by different dynamical regimes in the classical limit are identified with the presence of non-trivial minima in the separation of adjacent energy levels. Besides that, the properties of the ground state are used to characterize the order of the transition and also to construct a phase diagram that shares some similarity with the phase diagram obtained for the Lipkin model [25]. Our results can be applied to describe a system of exciton-polaritons in a semiconductor microcavity at the magic angle. This work is organized as follows. In Sec. II, the effective Hamiltonian is presented and, in Sec. III, its spectral properties are shown; in Sec. IV, the classical thermodynamic limit of the Hamiltonian is taken and the resulting phase space is analyzed. In Sec. V, we discuss the QPT and a phase diagram. In Sec. VI, we verify our results for exciton-polaritons in a semiconductor microcavity, based on available experimental data. Finally, we present our conclusions in Sec. VII.

II. THREE-MODE APPROXIMATION FOR INTERACTING BOSONS

The starting point of this work is a three-mode Hamiltonian for interacting bosons in the form

H = i=0,1,2 E i a † i a i + i+j=k+l G ijkl a † i a † j a k a l , (1)
where a † i (a i ) is the creation (annihilation) operator for a boson with energy E i . The first term describes the free bosons while the second one describes the interaction between different bosonic modes which obeys the condition: G ijkl = 0 for i + j = k + l, G ijkl = 0 for i + j = k + l. As we see in the following, we can obtain a Hamiltonian as Eq. [START_REF] Sachdev | Quantum Phase Transitions[END_REF] for the description of exciton-polaritons in a semiconductor microcavity at the magic angle (Sec. II.A). An effective Hamiltonian (Sec. II.B) can be obtained from Eq. (1) by using the conservation of total number of bosons and the condition i + j = k + l which means conservation of momentum for exciton-polaritons.

A. Exciton-polaritons in a semiconductor microcavity at the magic angle An exciton-polariton is a quasi-particle formed from the coupling between an exciton and a photon in a semiconductor microcavity. Under certain experimental conditions we can model the exciton-polaritons in a semiconductor microcavity by the Hamiltonian [START_REF] Kavokin | Microcavities[END_REF]11]

H = k Ω k p † k p k + 1 2 k,k ,q V P P k,k ,q p † k+q p † k -q p k p k , (2)
where p † k (p k ) is the creation (annihilation) operator for an exciton-polariton with in-plane wave-vector k and energy Ω k . The first term describes the free exciton-polaritons while the second one describes the interaction between different exciton-polariton modes. For typical values of the experimental parameters, the interaction coefficients are given by

V P P k,k',q V 0 u |k'-q| u k .u |k+q| u k , V 0 = 6e 2 a exc 0 A , (3)
where a exc is the two-dimensional Bohr radius of the exciton-polariton, 0 is the dieletric constant of the semiconductor and A is the macroscopic quantization area. The u k 's are the so-called Hopfield coefficients, given by

u k = ∆ k + ∆ 2 k + Ω 2 R 2 ∆ 2 k + Ω 2 R 1/2 , (4)
where

∆ k = E cav (k) -E exc (k)
is the detuning between the energies of cavity photons and excitons.

At the magic angle configuration, the system exhibits a parametric amplification of the emitted light [10]. In this situation, a theoretical description of the system is given by considering that only three modes for exciton-polaritons -namely, signal (0), pump (k p ), and idler (2k p ) -are coherently and macroscopicaly populated [11]. If k p is the wave vector of pumping, the scattering of two k p exciton-polaritons results in two other excitonpolaritons with wave vectors 0 and 2k p . Considering this dynamics, we can approximate Hamiltonian Eq. ( 2) by a three-mode Hamiltonian as Eq. ( 1) with p 0 ≡ a 0 , p kp ≡ a 1 , and

p 2kp ≡ a 2 .
In this way, the coefficients G ijkl are functions of the V P P k,k',q and the condition i + j = k + l in Eq. ( 1) is assured by the conservation of momentum in the second term of Eq. (2).

B. An effective Hamiltonian

The three-mode Hamiltonian Eq. ( 1) conserves the total number of bosons represented by

the observable N ≡ a † 0 a 0 + a † 1 a 1 + a † 2 a 2 .
Besides that, the difference between the population of bosons in modes 0 and 2, i.e., the imbalance represented by D ≡ a † 0 a 0 -a † 2 a 2 , is also conserved. These are two constants of motion under the evolution given by Eq. ( 1). In terms of N and D operators, we can rewrite Hamiltonian Eq. ( 1) as

H = H N D ( N , D) + H ef f (n 0 , a 0 a 2 a †2 1 ), (5)
where n0 ≡ a † 0 a 0 . For convenience and symmetry, we will consider N even and D = 0. We can observe that H N D is constant, giving rise only to a global phase in the state of the system as a function of time. The time evolution depends only on the second part of Eq. ( 5), an effective Hamiltonian given by

H eff = δn 0 + g(n 0 ) 2 + [Ga † 0 a † 2 a 2 1 + G * a 0 a 2 (a † 1 ) 2 ], (6)
where points. Moreover, we see that for g/δ = 0 the spectra are always symmetric with respect to the central level and both inflection points move accordingly toward the center as a function of G/δ. For values g/δ > 0, the spectra are asymmetric and the inflection points move separately toward the center as G/δ is varied. For g/δ < 0 [Fig. 2], we observe a more complex behavior with a two-fold degeneracy which disappears for G/δ > 0.0005 [Fig. 2 (a)], but in essence it is similar to the case g/δ > 0, with two asymmetrical inflection points moving separately to the centre of the spectrum (Fig. 2 (b)). When we calculate the mean values of population in mode 0, n 0 , and in mode 1, n 1 , for a degenerate spectrum, we find that, at a degeneracy, there is the possibility of population inversion between modes with the same energy value. In other words, two states with the same energy,

δ = E 0 + E 2 -2E 1 -G 0000 -G 2222 + 2 G 1111 + 4 N [G 1212 + G 0101 -G 1111 ], g = -8G 1212 + G 0000 + 4G 0202 -8G 0101 + 4G 1111 + G 2222 ,
E i = E i+1 , may have n 0 > n 1 or n 0 < n 1 .
Therefore, the degeneracy is characterized by the possibility of reversing the population of modes without altering the state's energy.

We can also observe the behavior of the spectrum in Fig. 3, where maximal level densities appear as minimal differences for adjacent energy levels. It has been shown [26,28] that in these Curie-Weiss models a quantum phase transition is connected with the level approximation, occurring maximally at these inflection points. We will also see that in the thermodynamic limit the inflection point is associated with a separatrix orbit in the corresponding classical phase space.

IV. CLASSICAL THERMODYNAMIC LIMIT OF THE HAMILTONIAN

The classical analog can be obtained as the thermodynamic limit of Hamiltonian Eq. ( 6).

The first step is the mapping of the operators into a SU(2) algebra by taking

J z ≡ n0 - N 4
, and With these operators, Hamiltonian Eq. ( 6) is rewritten as

J + = J † -= 1 2n 0 (N -2n 0 + 1) a † 0 a † 2 a 2 1 . (7)
H eff = δ(J z + J) + g(J z + J) 2 + G 2(J z + J)[N -2(J z + J) + 1]J + (9) +G * J -2(J z + J)[N -2(J z + J) + 1] with J = N/4.
The following step to the thermodynamic limit is obtained by rescaling the Hamiltonian by the density N/V and taking the limits N → ∞ and V → ∞, with the ratio N/V being kept constant. In this limit, the classical variables are provided by the usual definitions [30]

j k = lim J→∞ J k J , (k = +, -, z) (10)
and

j x = 1 2 (j + + j -) = 1 -j 2 z cos φ, (11)
where φ and j z correspond to canonical conjugate variables. The classical Hamiltonian finally obtained is written as

h(j z , φ) = δ (j z + 1) + g (j z + 1) 2 + 4G (1 -j 2 z ) cos(φ) (12)
with the rescaled parameters δ = δV /4, g = gN V /16 and G = GN V /16.

The classical phase space φ × j z is shown in Figs. 4-7, using, respectively, g /δ = 0, g /δ = 0.375, and g /δ = -0.375 for some given values of the three-mode coupling G /δ .

As expected, the phase space is periodic in the variable φ and j z is restricted to -1 ≤ j z ≤ 1.

In these figures we can see two different dynamical regimes: closed and open orbits separated by a separatrix corresponding to the classical energies for j z = ±1. The closed orbits correspond to classical energies less than h(j z = -1) or greater than h(j z = 1). The arising of the separatrix in phase space is associated to the level approximation in the quantum spectrum [26,28]. This can be seen in the three cases below:

(i) g /δ = 0 (g/δ = 0): We observe the appearance of two separatrices for G /δ > 0.125 (G/δ > 0.001) next to upper and lower classical energies h(j z = 1) and h(j z = -1), while in the quantum spectrum we observe the level approximation in both extreme energies.

(ii) g /δ = 0.375 (g/δ = 0.003): We observe the appearance of a separatrix for G /δ > 0.125 (G/δ > 0.001) next to the lower classical energy h(j z = -1) and for G /δ > 0.3125 (G/δ > 0.0025) next to the upper classical energy h(j z = 1), while in the quantum spectrum we observe respective level approximations next to upper and lower energies.

(iii) g /δ = -0.375 (g/δ = -0.003): Besides the behavior seen in (ii), we also observe a degeneracy of two classical trajectories with the same energy. This degeneracy disappears for G /δ > 0.0625 (G/δ > 0.0005). The previous aspects can be explained analytically by means of the critical points of the function h(φ, j z ). These are of two kinds:

-Maxima or minima localized at

(φ, j z ) max,min = 2nπ, δ + 2g 8G -2g (13)
and (φ, j z ) max,min = (2n + 1)π, -

δ + 2g 8G + 2g , (14)
-Saddle points at

(φ, j z ) saddle = arccos δ + 4g 8G , 1 (15)

and

(φ,

j z ) saddle = arccos -δ 8G , -1 . ( 16
)
These critical points are shown in Figs. 4567. The critical points in Eqs. ( 13) and ( 14) can be maxima or minima, depending on whether g /δ ≥ 0 or g /δ < 0. The closed orbits are created around the maximum and minimum points. Therefore, the conditions for the existence of these points are the conditions for the arising of the separatrix for upper and lower classical energies. We deduce from the conditions above that we have the arising of the two separatrices starting for the same value of the three-mode coupling G /δ = 0.125 when g /δ = 0. In the same way, when g /δ = 0.375 we have two distincts values of the

V. QUANTUM PHASE TRANSITION

We can observe characteristics of a quantum phase transition by looking at the properties of the ground state and of the measures of its entanglement [START_REF] Sachdev | Quantum Phase Transitions[END_REF]31]. In Fig. 8 we show the average value of the 0-mode population n 0 and the linear entropy S = 1 -T r[ρ 2 n 0 ] for the ground state of the system. Here, ρ n 0 stands for the reduced density operator of the 0-mode for the ground state, obtained by tracing out the other modes. We clearly observe two distinct behaviors for these quantities, for G/δ < 0.001 or G/δ > 0.001. This can be associated with the phase transition between the two dynamical regimes of macroscopic selftrapping (MST) and of oscillations (RO). For G/δ < 0.001, the average 0-mode population n 0 ] of the 0-mode as a function of the three-mode coupling G/δ for the ground-state of the system.

is practically null and we can deduce that the total population of bosons is on average in the 1-mode. This situation characterizes the MST phase in which the bosons remain in a single mode. This more organized phase has a relatively small linear entropy, so a small entanglement. For G/δ > 0.001, the average 0-mode population is non-null and increases with the three-mode coupling. In this situation we can say that the total population of bosons tends on average to be divided among the different modes. This situation characterizes the RO phase in which the bosons do not have a preferred mode and oscillate between them.

As the system is in a more disordered situation the linear entropy is larger, and so is the entanglement.

We can now discuss the order of the quantum phase transition. Following the usual criterion [START_REF] Sachdev | Quantum Phase Transitions[END_REF]31], we can obtain this information by analyzing the behavior of the ground state energy and the entanglement, and their derivatives. Such aspects, i.e., the second derivative of the energy and the first derivative of the entropy, both discontinuous for the ground state, characterize a second order phase transition. We note that a first-order transition would be characterized by a discontinuous first derivative of the ground state energy [START_REF] Sachdev | Quantum Phase Transitions[END_REF]31].

B. Phase diagram

We can now construct a phase diagram for the transition. We observe that in the quantum spectrum (Fig. 1) for G/δ < 0.001 the eigenenergies are restricted to values 0 < E < N/2 + (g/δ)(N/2) 2 , while for G/δ > 0.001 some eigenenergies E < 0 and E > N/2 + (g/δ)(N/2) 2

arise. These quantum states correspond precisely to classical closed orbits which arise for G/δ > 0.001. In this way we can associate these quantum states with the second phase of the system and separate the spectrum in different regions for the values E < 0 and

E > N/2 + (g/δ)(N/2) 2 .
Varying the three-mode coupling G/δ, we change the number of eigenstates (regions of the spectrum) associated with the second phase. While G/δ increases, the second phase region of the spectrum becomes larger. This phase diagram is shown in Fig. 11, where the vertical axis is a relative index i/i max of the eigenvalues E i . The lower curve separates the eigenvalues E i < 0 while the upper curve separates the eigenvalues

E i > N/2 + (g/δ)(N/2) 2 .
We can observe that for g/δ = 0 the arising of the second phase in upper energies is changed. A similar phase diagram had already been done for the Lipkin model in Ref. [25]. The difference between our three-mode model and the Lipkin model is just the asymmetry due to coefficient g/δ, since the Lipkin model has a symmetric phase diagram. This feature gives rise to a situation in this three-mode model which does not occur in the Lipkin model: the manifestation of the second phase in the upper energies is changed for g/δ = 0.

VI. APPLICATION

In the previous sections we have analyzed the behavior of the three-mode Hamiltonian and of its corresponding classical hamiltonian function for different values of the parameters G/δ and g/δ. But until now we have not discussed how we can vary physically these parameters.

We will examine below an example for exciton-polaritons in a semiconductor microcavity.

We can use experimental values of an exciton-polariton system at the magic angle configuration to test the results obtained in the previous sections. The values of the parameters G/δ and g/δ in the effective Hamiltonian Eq. ( 6) can be determined from the parameters Ω k and V P P k,k ,q in the exciton-polariton Hamiltonian Eq. ( 2). In Fig. 12 we observe a curve of G/δ as a function of the detuning ∆ 0 numerically calculated from experimental values [32]. In this figure we note that the increase in the three-mode coupling due to detuning variation results in phase transition from MST to RO. We emphasize that an exciton-polariton system is a very complex system with numerous features which were not considered in our simple analysis. For example, the coexistence of other system components such as photons, excitons, or bi-excitons, and the consequent conversion between them. Also, there is the non-conservation of total number of particles due to pumping and dissipation in the system. In this way, our work constitutes a first and simple approach to a complex system and is based on features of the macroscopic population of the three exciton-polariton modes at the magic angle configuration. In other words, we have an effective but physically meaningful model.

VII. CONCLUSIONS

The three-mode Hamiltonian studied in this work can be classified in a class of schematic models named Curie-Weiss models. There are many examples of this class as the Lipkin model and the pairing model (both in nuclear physics), the Dicke model for the superradiance, and others. Models in this class share the characteristic of a Hamiltonian which allows an expansion in powers of 1/N , which in a thermodynamic limit leads to a classical Hamiltonian analysis. In these models a quantum phase transition is signaled by a level approximation near an inflection point of the spectrum. This feature in the classical analysis is signaled by the appearance of a separatrix in the phase space. In the Curie-Weiss class, the Lipkin model has the particularity of a symmetric level approximation in both upper and lower inflection points of the spectrum. The three-mode Hamiltonian studied here also has this particularity and allows moreover for a situation (g/δ = 0) of an asymmetric level approximation in the upper and lower parts of the spectrum. In other words, we can have an asymmetric phase diagram.

The effective Hamiltonian obtained in this work can describe exciton-polaritons in a semiconductor microcavity at the magic angle. In terms of the exciton-polariton system, our results constitute an effective but physically meaningful model. It is a simple approach to this intricate system, based on the macroscopic features of the exciton-polariton population of only three modes.

  and G = 2G 1102 . In the next section we study the properties of the spectrum of the above Hamiltonian for different values of parameters G/δ and g/δ. III. QUANTUM SPECTRUM In Fig. 1 we observe the eigenvalues E i (i = 1, 2, ..., N/2+1) for the rescaled Hamiltonian E = H eff / δ as a function of the eigenstate index i for different values G/δ of the coupling between the three modes. We take g/δ = 0 [Fig. 1(a)], g/δ = 0.003 [Fig. 1 (b)], and g/δ = -0.003 [Fig. 2], and we choose N = 500. In both cases of Fig. 1, the spectra show maximal level densities at energies E = 0 and E = N/2 + (g/δ) (N/2) 2 , which are inflection

FIG. 1 .

 1 FIG. 1. Energy eigenvalues E i as a function of the eigenstate index i for g/δ = 0 (a), g/δ = 0.003 (b), N = 500, and different values of the three-mode coupling coefficient G/δ. The horizontal dotted lines mark the values E = 0 and E = N/2 + (g/δ)(N/2) 2 .

FIG. 2 .

 2 FIG. 2. Energy eigenvalues E i as a function of the eigenstate index i for g/δ = -0.003, N = 500, and different values of the three-mode coupling coefficient G/δ. For G/δ < 0.0005 (a) we observe a two-fold degeneracy which vanishes for G/δ > 0.0005 (b).

FIG. 3 .

 3 FIG. 3. Difference between adjacent energy eigenvalues E i+1 -E i as a function of the index i for g/δ = 0 (a), g/δ = 0.003 (b), and g/δ = -0.003 (c), N = 500, and different values of the three-mode coupling coefficient G/δ.

FIG. 4 .

 4 FIG. 4. Phase space φ × j z for g /δ = 0, G /δ = 0.125 (a), and G /δ = 0.25 (b). Rescaled parameters calculated for N = 500, g/δ = 0, G/δ = 0.001 (a), and G/δ = 0.002 (b). The critical points of h(φ, j z ) are signaled by black dots.

FIG. 5 .

 5 FIG. 5. Phase space φ × j z for g /δ = 0.375, G /δ = 0.3125 (a), and G /δ = 0.375 (b). Rescaled parameters calculated for N = 500, g/δ = 0.003, G/δ = 0.0025 (a), and G/δ = 0.003 (b). The critical points of h(φ, j z ) are signaled by black dots.

FIG. 6 .

 6 FIG. 6. Phase space φ × j z for g /δ = -0.375, G /δ = 0.0125 (a), and G /δ = 0.0625 (b). Rescaled parameters calculated for N = 500, g/δ = -0.003, G/δ = 0.0004 (a), and G/δ = 0.0005 (b). The critical points of h(φ, j z ) are signaled by black dots and the degenerated open orbits by trace-dotted curves.

FIG. 7 .

 7 FIG. 7. Phase space φ × j z for g /δ = -0.375, G /δ = 0.125 (a), and G /δ = 0.15 (b). Rescaled parameters calculated for N = 500, g/δ = -0.003, G/δ = 0.001 (a), and G/δ = 0.002 (b). The critical points of h(φ, j z ) are signaled by black dots.

FIG. 8 .

 8 FIG. 8. Average value of population n 0 (a) and the linear entropy S = 1 -Tr[ρ 2n 0 ] of the 0-mode

Figure 9

 9 figure we observe the minimum of the corresponding classical energy h(j z , φ), which shows a perfect agreement with the ground state energy rescaled as E/(N/4). Taking the first and second derivatives, we observe that the first derivative is continuous and the second one is discontinuous [Fig.9 (b)]. In Fig.10the first derivative of the linear entropy as a function of the three-mode coupling G /δ is shown for different values of the total boson population N . We observe that the curves tend to diverge as N → ∞ at the critical value G /δ = 1.25.

FIG. 9 .

 9 FIG.9. The ground state energy (a) and its second derivative (b) as a function of the three-mode coupling G /δ . In (a) we also observe the ground state energy rescaled as E/(N/4) (circle).

FIG. 10 .FIG. 11 .

 1011 FIG. 10. The first derivative of the linear entropy S = 1 -T r[ρ 2 n 0 ] as a function of the three-mode coupling G /δ for the ground state and different total boson populations N = 200, 500, and 1000.

FIG. 12 .

 12 FIG.12. Numerical value for G/δ as a function of the detuning ∆ 0 , calculated from experimental parameters of a semiconductor microcavity[32]. The horizontal dotted line marks the critical value G/δ = 0.00005 for N = 10 4 .

ACKNOWLEDGMENTS

We would like to thank CNPq (Grant No. 312207/2015-8) and CAPES for financial support. H.M.F., J.G.P.F. and G.Q.P. would like to dedicate this work to the memory of Maria Carolina Nemes.


Keywords: random environment, random walk, Sinai's regime, system of particles, excursion

A limit result for a system of particles in random environment Pierre Andreoletti *

August 28, 2007

Abstract: We consider an infinite system of particles in one dimension, each particle performs independant Sinai's random walk in random environment. Considering an instant t, large enough, we prove a result in probability showing that the particles are trapped in the neighborhood of well defined points of the lattice depending on the random environment the time t and the starting point of the particles.

Introduction, definitions and result

Systems of particles have been study a lot in many different directions see for example [START_REF] Ligget | Interacting Particle systems[END_REF], [START_REF] Demasi | Mathematical Methods for Hydrodynamic Limits[END_REF] and [START_REF] Kipnis | Scaling limits of interracting particle systems[END_REF] and the references therein. In this paper we are interested in a system of particles performing independant Sinai's random walk in random environment on the latice Z. At time 0, in each point of the lattice i ∈ Z a random number of particles η(i, 0) is distributed, we assume that the sequence (η(i, 0), i ∈ Z) is i.i.d. and that η(0, 0) is distributed according to a Poisson distribution of parameter λ. Each particle of this system effectuates a Sinai's walk that means a one dimensional random walk in random environment with three conditions on the random environment: two necessaries hypothesis to get a recurrent process (see [START_REF] Solomon | Random walks in random environment[END_REF]) which is not a simple random walk and an hypothesis of regularity which allows us to have a good control on the fluctuations of the random environment. The asymptotic behavior of such a walk has been understood by [START_REF] Ya | The limit behaviour of a one-dimensional random walk in a random medium[END_REF] : it is sub-diffusive, that is, for an instant t it behaves like (log t) 2 and it is localized in the neighborhood of a well defined point of the lattice. We are interested in the general behavior of the system described above in the following sense, let us denote η(x, t) the number of particles at time t on x, and let f be a differentiable function with compact support, we study the asymptotic behavior when t becomes large of 1 (log t) 2 x∈Z η(x, t)f (x/(log t) 2 ) .

(1.1)

The result we get can been seen as the generalization of Sinai's localization result for a system of particles. It shows that the particles are trapped in the bottom of the deepest valley that the walk can reach within an amount of time t. The coordinates of the bottom of the valleys depend on the starting points of the particles, the random environment and t.

To get the result, first, we make a construction on the random environment closed to the one that [START_REF] Neveu | Renewal property of the extrema and tree property of the excursion of a one-dimensional Brownian motion[END_REF] use to study the excursion of the Brownian motion, thanks to this we get a partition in term of valleys of the part of the lattice related to the support of f . Then we prove that there is neither importation of particles from outside of the support of the function f nor exchanged of particles between the (deepest) valleys, finally we we use the accurate result of [START_REF] Andreoletti | Alternative proof for the localisation of Sinai's walk[END_REF] about Sinai's localization result extended to a finite number of particles.

In the following section, we define Sinai's walk, then the system of particles and finally, we state the main result. In section 2 we make the construction on the random environment, in section 3 and 4 we prove the main theorem.

Definition of Sinai's walk

Let α = (α i , i ∈ Z) be a sequence of independent and identically distributed (i.i.d.) random variables taking values in (0, 1) defined on the probability space (Ω 1 , F 1 , Q), this sequence will be called random environment. A random walk in random environment (denoted R.W.R.E.) (X n , n ∈ N) is a sequence of random variable taking value in Z, defined on (Ω, F, P) such that • for every fixed environment α, (X n , n ∈ N) is a Markov chain with the following transition probabilities, for all n ≥ 1 and i ∈

Z P α,1 [X n = i + 1|X n-1 = i] = α i , (1.2) P α,1 [X n = i -1|X n-1 = i] = 1 -α i .
We denote (Ω 2 , F 2 , P α,1 ) the probability space associated to this Markov chain.

• Ω = Ω 1 × Ω 2 , ∀A 1 ∈ F 1 and ∀A 2 ∈ F 2 , P [A 1 × A 2 ] = A 1 Q(dw 1 ) A 2 P α(w 1 ),1 (dw 2 ).
The probability measure P α,1 [ .| X 0 = a] will be denoted P α,1 a [.], the expectation associated to P α,1 a : E α,1 a , and the expectation associated to Q: E Q .

Now we introduce the hypothesis we will use in all this work. The two following hypothesis are the necessaries hypothesis [START_REF] Solomon | Random walks in random environment[END_REF] shows that under 1.3 the process (X n , n ∈ N) is P almost surely recurrent and 1.4 implies that the model is not reduced to the simple random walk. In addition to 1.3 and 1.4 we will consider the following hypothesis of regularity, there exists 0

E Q log 1 -α 0 α 0 = 0, (1.3) Var Q log 1 -α 0 α 0 ≡ σ 2 > 0. (1.4)
< ρ 0 < 1/2 such that sup {x, Q [α 0 ≥ x] = 1} ≥ ρ 0 and sup {x, Q [α 0 ≤ 1 -x] = 1} ≥ ρ 0 . (1.5)
We call Sinai's random walk the random walk in random environment previously defined with the three hypothesis 1.3, 1.4 and 1.5. Instead of considering the discrete time process, we will work with the continuous time one (X t , t ∈ R + ) define as follows, ∀t ∈ R + , X t = X [t] where [t] is the integer par of t.

The system of particles

We recall that η(i) ≡ η(i, 0) is the number of particles in i ∈ Z at time 0, we assume that (η(i), i ∈ Z) is i.i.d, and that η(0) is distributed according to a Poisson distribution of parameter λ, we denote P 1 the corresponding probability measure. We denote X x,i t the coordinate of a particle at time t corresponding to the i th particle, which was on site x at time 0 (X x,i 0 = x). η(x, t) can be written in the following useful way:

η(x, t) = y∈Z η(y) i=1 I {X x,i t =x} (1.6)
The distribution of the whole system is denoted P , it is the product of Q and P α , where P α is the product

P α ≡ x∈Z P α x = P 1 × x∈Z η(x) i=1 P α,i
x . The measure P generalizes the measure P defined Section 1.1 for a random number of particles. Here we have three levels of randomness, the environment, the number of particles per sites and the random walks themselves, hence both P α and P α

x are random measures. Notice that, when the environment is fixed (η(., t)) inherits the markov property from the random walks, indeed they all have the same transition probability and they do not interact with each other.

Main results

First let us state the result in probability, the following Theorem is based on a construction on the random environment (see figure 3). Let C 1 κ be the set of all differentiable functions with a compact support.

Theorem 1.1. Assume 1.3, 1.4 and 1.5 hold, for all f ∈ C 1 κ , and all t large enough

P   1 (log t) 2 x∈Z η(x, t)f (x/(log n) 2 ) -λ n(f ) i=1 |M i+1 -M i | (log t) 2 f (m i /(log t) 2 ) = o(1)   = 1 -o(1),
where, (m i , i ≥ 1), (M i , i ≥ 1) and n(f ) are well defined variables depending only on the random environment α, on the time t and f . o(1) is a positive decreasing function such that lim t→+∞ o(1) = 0. This Theorem says that at time t the particles are localized in the neighborhood of well defined points of the lattice. These points have the coordinates of the bottoms of the valleys where the particles started their walk. In other words it says that the number of particles at time t, at the points m i is equal to λ|M i+1 -M i | which is the total number of particles at the instant t = 0 present in the valley ({M i , m i , M i+1 }). n(f ) is the number of valleys deep enough within the support of f , notice that n(f ) can be equal to zero if the support of f is too small. The following Proposition give the limit distributions of the random variables present in the Theorem.

Proposition 1.2. Assume 1.3, 1.4 and 1.5 hold, for all i ∈ N * , in distribution when t goes to infinity

σ 2 M i (log t) 2 → L i , (1.7) σ 2 m i (log t) 2 → L i , (1.8)
The (L i+1 -L i , i ∈ Z) are independent and equidistributed random variables with Laplace transform given by E(exp(-λ(L 2 -L 1 ))) = 1/(cosh( (2λ))) 2 , for all λ > 0. Moreover when the support of f is large enough the distribution of n(f ) is given by a normal law with mean σ 2 supp(f )/2 and variance 3 * σ 4 supp(f )/4.

Construction for the random environment

In this section we begin with some basic notions on the random environment of Sinai's random walk, then we construct the key random variables (M i , m i , i ≤ n(f )) appearing in our result. The reader can follow the different steps of the construction with the Figures 2345.

Definition and basic notions of valleys

For completeness we begin with some basic notions originally introduced by Sinai [1982]. The random potential and the valleys Let

ǫ i ≡ log 1 -α i α i , i ∈ Z, (2.1) define :
Definition 2.1. The random potential (S k , k ∈ R) associated to the random environment α is defined in the following way:

S k = 1≤i≤k ǫ i , if k = 1, 2, • • • , -k+1≤i≤0 ǫ i , if k = -1, -2, • • • , S 0 = 0.
for the other k ∈ R \ Z, S k is defined by linear interpolation.

Definition 2.2. We will say that the triplet {M ′ , m, M ′′ } is a valley if

S M ′ = max M ′ ≤t≤m S t , (2.2) S M ′′ = max m≤t≤ M ′′ S t , (2.3) S m = min M ′ ≤t≤M ′′ S t . (2.4)
If m is not unique we choose the one with the smallest absolute value.

Definition 2.3. We will call depth of the valley {M ′ , m, M ′′ } and we will denote it d([M ′ , M ′′ ]) the quantity min(S M ′ -S m , S M ′′ -S m ).

(2.5) Now we define the operation of refinement Definition 2.4. Let {M ′ , m, M ′′ } be a valley and let M 1 and m 1 be such that m ≤ M 1 < m 1 ≤ M ′′ and

S M 1 -S m 1 = max m≤t ′ ≤t ′′ ≤M ′′ (S t ′ -S t ′′ ). (2.6)
We say that the couple (m 1 , M 1 ) is obtained by a right refinement of {M ′ , m, M ′′ }. If the couple (m 1 , M 1 ) is not unique, we will take the one such that m 1 and M 1 have the smallest absolute value. In a similar way we define the left refinement operation.

We denote log 2 ≡ log log, in all this work we will suppose that t is large enough such that log 2 t is positive.

Definition 2.5. Let γ > 0, define Γ t ≡ log t + γ log 2 t, we say that a valley {M ′ , m, M ′′ } is of depth larger than Γ t if and only if d

([M ′ , M ′′ ]) ≥ Γ t .
2.2 Construction of a random cover for (supp(f )) * (log t) 2

We start with the constructions of a sequence of valleys with identical properties, the different steps of the construction are shown in Figures 1 to 2. Notice that the variables in greek letters we use below are not the fundamental one, they just help us to construct m + . , m - . , M + . , M - . and

M 0 . Let u, v ∈ R, u < v, define S - u,v = min{S s , u ≤ s ≤ v}, (2.7) S + u,v = max{S s , u ≤ s ≤ v}, (2.8) τ + 0 = inf{s > 0, S s -S - 0,s ≥ Γ t }, (2.9) m + 0 = sup{s < τ + 0 , S s = S - 0,τ + 0 }, (2.10) σ + 0 = inf{s > m + 0 , S + m + 0 ,s -S s ≥ Γ t },
(2.11)

τ - 0 = sup{s < 0, S s -S - s,0 ≥ Γ t }, (2.12) m - 0 = sup{s > τ - 0 , S s = S - 0,τ + 0 }, (2.13) σ - 0 = sup{s < m - 0 , S + m - 0 ,s -S s ≥ Γ t },
(2.14)

M 0 = inf{s > m - 0 , S s = S + m - 0 ,m + 0 }. (2.15) M 0 m + 0 τ - 0 τ + 0 (S k , k ∈ R) = Γt m - 0 0 σ - 0 σ + 0 Figure 1: Construction of the valleys 1/2
Now define recursively the following variables, let i ≥ 1

τ + i = inf{s > σ + i-1 , S s -S + σ + i-1 ,s ≥ Γ t },
(2.16)

m + i = sup{s > σ + i-1 , S s = S + σ + i-1 ,τ + i }, (2.17) M + i = inf{s > m + i-1 , S s = S + m + i-1 ,m + i }, (2.18) σ + i = inf{s > m + i , S + m + i ,s -S s ≥ Γ t }, (2.19)
and in the same way on the left hand side of the origin. We would like to make remarks about this construction:

Remark 2.6. Notice that for all i ≥ 1 the {M + i , m + i , M + i+1 } are valleys of depth larger than Γ t such that we can not construct a sub valley (with the operation of refinement, see Definition 2.4 ) of depth larger than Γ t , the same remark is true if we replace the + by the -. However we do not know how

(S k , k ∈ R) m + 0 τ - 0 τ + 0 m - 0 0 σ - 0 = Γt σ + 0 τ + 1 τ + 2 σ + 2 M + 3 m + 2 M + 2 M + 1 m + 1 σ + 1 M - 1 M 0 Figure 2: Construction of the valleys 2/2 deep the valleys {M - 1 , m - 0 , M 0 } and {M 0 , m + 0 , M + 1 }
are, what we know from our construction is that we must have one of these two cases:

min d [M - 1 , M 0 ] , d [M 0 , M + 1 ] ≥ Γ t or (2.20) min d [M - 1 , M 0 ] , d [M 0 , M + 1 ] < Γ t and d [M - 1 , M + 1 ] ≥ Γ t (2.21)
In our drawings it is the case 2.20 that occurs. Now let us describe the link between the construction we have exposed above and the support of f . For simplicity, we will assume that supp(f ) = [-K, K] with K > 0. Let n + be the largest integer such that m n + < K(log t) 2 and symmetrically n -the largest integer such that m n -> -K(log t) 2 . Note that both n + and n -can be equal to zero. It is also important to notice that, we can have 2 different cases in both on the right and left hand-side of the origin. We will only discuss about the right hand-side: first case is when m n + < K(log t) 2 < M n + +1 in this case we will have to consider the particles that belong to the interval (K(log t) 2 , M n + +1 ). Indeed most of them will tend to reach the bottom the valley m n + that belongs to [-K(log t) 2 , K(log t) 2 ]. To the contrary in the case where M n(f )+1 > K(log t) 2 > m n + +1 , we will not have to care about the particles in the interval (M n(f )+1 , K(log t) 2 ) because they will all tend to reach m n + +1 that do not belongs to [-K(log t) 2 , K(log t) 2 ].

To simplify notations we re-numerate all the set of minima and maxima in the following way : we suppress the "+" and the "-", so denoting n(f

) ≡ n(K) = (n + + n --I A c t ) ∨ 0 where A t = {min d [M - 1 , M 0 ] , d [M 0 , M + 1 ] ≥ Γ t }, we denote: M - n -= M 1 , m - n -= m 1 , M - n --1 = M 2 , ..., M + n + = M n(f )+1
. Note that the -I A c t is needed because we have two cases (2.20 and 2.21). In our drawing (see Figure 3) n + = 2, n -= 1 and we are in the case 2.20 so n(f ) = 3.

Indeterminate points and Set of good environments

Let x ∈ [-K(log t) 2 , K(log t) 2 ], due to the preceding construction we know that there exists

i ∈ N such that x ∈ [M i , M i+1 ], let ī ≡ ī(t, f, α) = {y ∈ Z, y ∈ [M i , M i+1 ]}, i ≤ n(f ). Now let us define the set of indeterminate points. Let x ∈ l, 1 ≤ l ≤ n(f ) + 1, we will say that x is indeterminate if x ∈ U l ≡ [M l -(log 2 t) 2 , M l + (log 2 t) 2 ].
We call these sets "indeterminate", because at each top of the valley, actually a small interval on the top, we can not determine which valley the particles starting from there will choose. Notice, that the number of indeterminate points, belonging to V f , are negligible comparing to a typical fluctuation of Sinai's random walk.

M 4 m 2 (S k , k ∈ R) m 1 0 = Γt M 5 m 4 M 3 m 3 M 1 M 2 K(log t) 2 -K(log t) 2
U 1 m 2 m 1 0 M 5 m 4 m 3 K(log t) 2 -K(log t) 2 (S k , k ∈ R) 2 M 1 M 2 M 3 M 4 1 3 U 2 U 3 U 4
n(f ) is finite, (2.22) |V f | ≤ c 1 (log n) 2 , (2.23) ∀1 ≤ i ≤ n(f ), | ī| ≡ |M i -M i+1 | ≈ c 2 (log n) 2 , (2.24) ∀1 ≤ i ≤ n(f ) the valley{M i , m i , M i+1 } have Sinai's basic properties. (2.25)
Remark 2.7. (about 2.25) As we said before to get our result we need to prove the result of localization of Sinai for independent particles. For that we need some properties on the random environment we do not want to repeat here (see for example [START_REF] Andreoletti | Alternative proof for the localisation of Sinai's walk[END_REF] Definition 3.4, page 894), we have called these properties "Sinai's basic properties".

Let us define the set of good properties G t , G t = {α ∈ Ω 1 , α satisfies 2.22 -2.25}. The following Proposition, give a probability result for G t : Proposition 2.8. There exists c 1 > 0, c 2 > 0 such that for all t large enough, Q[G t ] = 1 -o(1).

Proof.

We shortly discuss the proof of this Proposition at the end of the paper, Paragraph 5.

Results of localization for the particles

In this section we first prove two results (Proposition 3.1), about the no-migration of the particles from the valley where they start to another (different) one, then we state the key result which generalizes Sinai's result for a system of independent particles, let f ∈ C 1 κ we denote

V f = { 1, • • • , n(f )} the random cover of [-K(log t) 2 , K(log t) 2 ]
(see the construction in the previous paragraph), we have Proposition 3.1. For all f ∈ C 1 κ , γ > 12, and all t large enough,

Q[G t ] = 1 -o(1) and inf α∈Gt    P α   x / ∈V f η(x) k=1 I X x,k t ∈V f = o(log 2 t)      = 1 -o(1), (3.1)
conversly, we have:

inf α∈Gt    P α   n(f ) i=1 x∈ ī U i η(x) k=1 X x,k t ∈ ī      = 1 -o(1). (3.2)
3.1 shows that with an overwhelming probability a negligible number of particles, coming from the complementary of V (f ) in Z, will reach V f before the instant t. 3.2 shows that, within the interval of time t, there is no exchange of particles between two valleys except, may be, for the particles that belong to the sets of indeterminate points. Proof. First let us give the proof of 3.2, it is based on the two following facts,

P α   n(f ) i=1 x∈ ī U i η(x) k=1 X x,k t ∈ ī   ≥ exp   λ [a -1] n(f ) k=1 | k U k |   , ∀α, (3.3) a ≥ 1 - cte (log t) γ-2 , ∀α ∈ G t . (3.4) where a = inf 1≤i≤n(f ) inf x∈ ī U j P α x X i,1
t ∈ ī . Fact 3.4 is basic, to get it we use a similar method that [START_REF] Andreoletti | Alternative proof for the localisation of Sinai's walk[END_REF] use to get the Proposition 2.8 page 888. We now give a proof of Fact 3.3,

P α   n(f ) i=1 x∈ ī U i η(x) k=1 X x,k t ∈ ī   = P α   n(f ) i=1 x∈ ī U i +∞ kx=0 η(x) = k x , kx k=1 X x,k t ∈ ī   , (3.5) = n(f ) i=1 x∈ ī U i P α x   +∞ kx=0 η(x) = k x , kx k=1 X x,k t ∈ ī   , = n(f ) i=1 x∈ ī U i exp λ(P α,1 x X x,1 t ∈ ī -1) .
In the first equality we have introduced a partition over the values of the (η(x), x ∈ Z), the second equality comes from the independence of the particles starting from distinct points of the lattice, the last one from the fact that (X .,s . , s ∈ N) are i.i.d. Finally

P α   n(f ) i=1 x∈ ī U i η(x) k=1 X x,k t ∈ ī   ≥ n(f ) i=1 y∈ ī U i
exp (λ(a -1)) , ∀α, (3.6) so we get 3.3. To end the proof we use 2.23 and 2.24. Notice that Fact 1 is true for all environments α whereas Fact 2 is only true for all good environments. To get 3.1 we will consider three different distances from the starting point of the particles to the boundary of V f : M 1 or M n(f )+1 (Cases 1-3 below). We will only discuss the case when the boundary is M n(f )+1 , the other case can be treated in the same way. Case 1 (very long distance) notice that ∀x

> M n(f )+1 + t, X x,. t / ∈ V f , Case 2 (long distance), let x ∈ J t ≡ [M n(f )+1 +log 2 t(log t) 2 , M n(f )+1 +t].
With the same computations we have obtained 3.3 we can prove that

P α   x∈Jt η(x) k=1 {X x,k t / ∈ V f }   ≡ P α   x∈Jt η(x) k=1 {X x,k t > M n(f )+1 }   ≥ exp (tλ(b -1)) , ∀α (3.7) where b = inf x∈Jt P α x X t > M n(f )+1 , moreover it is a basic fact (see Andreoletti [2005] Proposition 2.8 page 888) that for all α ∈ G t , b ≥ 1 -cte/(t(log t) γ-2 ). Case 3 (short distance), let x ∈ K t ≡ [M n(f )+1 + (log 2 t) 2 , M n(f )+1 + (log 2 t)(log t) 2 ], we have P α   x∈Kt η(x) k=1 {X x,k t / ∈ V f }   ≥ exp (log 2 t)(log t) 2 λ(c -1) , ∀α, (3.8) where c = inf x∈Kt P α,1 x X t > M n(f )+1
, moreover it is also a basic fact, see the reference above, that for all α ∈ G t , c ≥ 1 -cte/(log t) γ-2 . From this three cases we deduce that the only particles, starting from a point inside (M n(f ) + 1, +∞), that can reach M n(f ) + 1, are the particles that belong to the interval (M n(f )+1 , M n(f )+1 + (log 2 t) 2 ), which is a subset of the set of indeterminate points U Mn(f )+1 , of size (log 2 t) 2 , negligible comparing to (log n) 2 . Proposition 3.2. For all f ∈ C 1 κ , γ > 12, and all t large enough, Q[G t ] = 1 -o(1) and

inf α∈Gt    P α   n(f ) i=1 x∈ ī U i η(x) k=1 X x,k t (log t) 2 -m i (t) ≤ cte (log t) 1/2      = 1 -o(1). (3.9)
The Proposition above generalizes Sinai's localization result for independent particles. It shows that each particle from a given valley will be located at time t in a small neighborhood of the coordinate of the bottom of this same valley.

Proof.

The proof of this Proposition is based on the two following facts, let I t,i = (m i (t)-cte(log t) 3/2 , m i (t)+ cte(log t) 3/2 ), we have

P α   n(f ) i=1 x∈ ī U i η(x) k=1 X x,k t ∈ I t,i   ≥ exp   λ [d -1] n(f ) k=1 | k U k |   where d = inf 1≤i≤n(f ) inf x∈ ī U i P α,1 i X x,1 t ∈ I t,i
, which is obtained with the same computation we did to get the previous Proposition, and

d ≥ 1 - cte (log n) γ-12 (3.10)
which is obtained by using a similar method of [START_REF] Andreoletti | Alternative proof for the localisation of Sinai's walk[END_REF] (Theorem 2.11 page 889).

Proof of Theorem 1.1

In all this section we will always assume that α ∈ G t . The two Propositions 3.1 and 3.2 will be used frequently, as these results are true in P α probability we will mention it by using " =" instead of the common "=". First using the definition of η(x, t) and 3.1, we get

y∈Z η(y, t)f (y/(log t) 2 ) ≡ x∈Z η(x) i=1 y∈Z I {X x,i t =y} f (y/(log t) 2 ) (4.1) = n(f ) l=1 x∈ l η(x) k=1 y∈Z I {X x,k t =y} f (y/(log t) 2 ) + o(log 2 t). (4.2)
Extracting the indeterminate particles and using 3.2, we have

n(f ) l=1 x∈ l η(x) k=1 y∈Z I {X x,k t =y} f (y/(log t) 2 ) ≡ n(f ) l=1 x∈ l-U l η(x) k=1 y∈Z I {X x,k t =y} f (y/(log t) 2 ) + ǫ (1) n = n(f ) l=1 x∈ l-U l η(x) k=1 y∈ l I {X x,k t =y} f (y/(log t) 2 ) + ǫ (1) n , (4.3) where ǫ (1) n = n(f ) l=1 x∈U l η(x) k=1 y∈Z I {X x,k t =y} f (y/(log t) 2
). Thanks to the fact that f ∈ C 1 κ , the law of large number for the sum of η(x) and that by definition |U . | = 2(log 2 t) 2 , it is easy to see that in

P 1 probability ǫ (1) n = O(n(f )(log 2 t) 2 ). Proposition 3.2 yields n(f ) l=1 x∈ l-U l η(x) k=1 y∈ l I {X x,k t =y} f (y/(log t) 2 ) = n(f ) l=1 x∈ l-U l η(x)
k=1 y∈I(t,l)

I {X x,k t =y} f (y/(log t) 2 ) (4.4)
where I(t, l) = {m l (t) -cte(log t) 3/2 , ..., m l (t) + cte(log t) 3/2 }. Now using the fact that f ∈ C 1 κ , we get that f (m l /(log t) 2 ) + O 1/(log t) 1/2 l -U l + o(n(f ) log 2 t).

To end the proof we use 2.22, the fact that by definition |U . | = 2(log 2 t) 2 , and we divide what we get by (log t) 2 .

5 Proof of Propositions 1.2 and 2.8

Proof of Proposition 1.2 Using the vocabulary of [START_REF] Neveu | Renewal property of the extrema and tree property of the excursion of a one-dimensional Brownian motion[END_REF][START_REF] Neveu | Renewal property of the extrema and tree property of the excursion of a one-dimensional Brownian motion[END_REF][1989b], the points M i , and m i are Γ t -extrema for a random walk. Moreover thanks to the theorem of Donsker (see for example [START_REF] Durrett | Probability: Theory an Examples[END_REF], page 406) (S tl / √ l, l) converge in distribution to a two-sided Brownian motion, therefore we get that the points L i are 1-minima and S i 1-maxima for the two sided Brownian motion. The first part of our Proposition 1.2 follows from the Proposition page 241 of [START_REF] Neveu | Renewal property of the extrema and tree property of the excursion of a one-dimensional Brownian motion[END_REF], indeed the 1-extrema built a stationary renewal process, such that the difference of two consecutive extrema form a i.i.d. sequence with Laplace transform equal to 1/ cosh( (2λ)). Moreover n(f ) is the number of renewal within the support of f . It is easy to check that P[n(f

) ≤ k] = P[R k ≤ |supp(f )|],
where R k = k l=1 x l with x l = L l+1 -L l so we get that if supp(f ) is large the distribution of n(f ) is given by a normal law with mean σ 2 supp(f )/2 and variance σ 4 supp(f ) * 3/4. Sketch of the proof of Proposition 1.2 The first 3 properties can easily be deduced from the proof above. About Sinai's basic properties the proof can be found in the Appendix A of [START_REF] Andreoletti | Alternative proof for the localisation of Sinai's walk[END_REF].

Figure 3 :

 3 Figure 3: Support of f and the valleys

Figure 4 :

 4 Figure 4: Random cover and sets of indeterminate points with n(f ) = 3.

II

  {X x,k t =y} f (y/(log t) l /(log t) 2 ) + O 1/(log t) {X x,k t =y} f (y/(log t) 2 ) = n(f ) l=1 f (m l /(log t) 2 ) + O 1/(log t) 1/2 x∈ l-U l η(x),collecting what we did above and using once again the law of large number, we get y∈Z η(y, t)f (y/(log t) 2 ) =λ n(f ) l=1

Acknowledgment I would like to thank Errico Presutti for introducing me the subject and for several very helpful discussions.