∎

11institutetext: M. Zohar 22institutetext: M. Auslender 33institutetext: S.
Hava 44institutetext: Department of Electrical and Computer Engineering, Ben
Gurion University of the Negev, P.O. Box 653, 84105 Beer Sheva
Tel.:+972-8-6461583
Fax: +972-8-6472949
44email<EMAIL_ADDRESS>55institutetext: L. Faraone 66institutetext:
School of Electrical, Electronic and Computer Engineering, The University of
Western Australia, M018 35 Stirling Highway Crawley WA 6009, Australia

# New resonant cavity-enhanced absorber structures for mid-infrared detector
applications

Moshe Zohar  Mark Auslender  Lorenzo Faraone  Shlomo Hava

(Received: date / Accepted: date)

###### Abstract

A new dielectric Fabry-Perot cavity was designed for a resonant enhancing
optical absorption by a thin absorber layer embedded into the cavity. In this
cavity, the front mirror is a subwavelength grating with $\sim 100$%
retroreflection. For a HgCdTe absorber in a matching cavity of the new type,
the design is shown to meet the combined challenges of increasing the
absorbing efficiency of the entire device up to $\sim 100$ % and reducing its
size and overall complexity, compared to a conventional resonant cavity
enhanced HgCdTe absorber, while maintaining a fairly good tolerance against
the grating’s fabrication errors.

###### Keywords:

Optical resonant cavity Photodetectors HgCdTe Gratings

##  1 Introduction

It is well known, that incorporating a photosensitive or an optically active
layer into a Fabry-Perot (FP) cavity enhances the efficiency of detection by
or emission from the layer, as a consequence of the multiple reflections that
occur between the cavity’s mirrors. The enhancement is maximized if the round-
trip phase, i.e. the phase difference between each succeeding reflection,
satisfies the resonance condition

$\delta_{0}\equiv\delta\left(\lambda_{0}\right)=\frac{4\pi n_{\rm
c}\left(\lambda_{0}\right)t_{c}}{\lambda_{0}}+\varphi_{\rm
f}\left(\lambda_{0}\right)+\varphi_{\rm b}\left(\lambda_{0}\right)=2\pi n.$
(1)

Here $\lambda_{0}$ is a resonance wavelength $n_{\rm c}$ and $t_{\rm c}$ is
the refractive index (RI) and length of the FP cavity, respectively,
$\varphi_{\rm f}$ and $\varphi_{\rm b}$ are the reflection phase of the mirror
that is further away from and adjacent to the illuminated side (hereafter
referred to as the front and back mirror, respectively), and $n$ is an
integer. Eq.(1) is thumb rule accurate regardless of the type of flat-surface
mirror being employed, whether it consist of thin metallic films, or is e.g. a
distributed Bragg reflector (DBR) that is a stack of quarter-wave pairs of
high/low (HL) RI dielectric layers heavens ; knittl .

Optical communication, interconnection and information processing systems
require high-efficiency photodetectors (PDs). In response to this demand, the
field of resonant-cavity-enhanced (RCE) PDs has steadily maturated over past
two decades unlu ; deen . For thermal PDs, the optical absorbance $A$ is an
accurate measure of the efficiency, while for photodiodes and photo-conductors
the key figure is the quantum efficiency $\eta$. For initial design of the RCE
PDs, the adoption of $\eta\approx A$ has a widespread use unlu ; deen  since
thus the analysis is framed into optics alone rather than being restricted to
a specific PD technology. It appears unlu  that for obtaining $\eta\approx
100\%$ it is at least necessary to have $\sim 100$% resonant reflectivity from
the front mirror, which is problematic for a DBR of any practical thickness
because of the tight material requirements for epitaxial growth. Thus, it is
relevant to consider an alternative replacement for the DBR mirror by a thin
dielectric micro-optical component which can be $\sim 100$ % reflective in a
broad spectral band. A device that can exhibit such a reflection anomaly with
a proper design havaus ; chang-a  is a dielectric subwavelength grating
structure.

This paper presents a theoretical study to incorporate a subwavelength grating
as the front mirror in the FP cavity, for PD applications in the mid-wave
infrared (MWIR) range. For a vertical-cavity surface-emitting laser, using an
air-bridge subwavelength grating as a suspended top mirror, has been proposed
before bissaillon ; kim . Only limited work on RCE PDs for operation in the
MWIR range has been reported thus far. Currently this is an area of great
interest due to the potential for obtaining the uncooled PDs.

##  2 Design considerations and simulation tools

We adopted $\eta=A$ [see Sec. 1], so RCE PDs with an absorber layer
characterized by a complex RI, $n_{\rm a}+ik_{\rm a}$, were analyzed only
optically. To compute the reflectance, transmittance and $A$ spectra of RCE
PDs including a grating, we used the rigorous coupled wave analysis recast
with an in-layer S-matrix propagation algorithm aushava1 . For DBR based PDs,
a limiting case at zero grating groove width, which coincides with the
impedance form knittl  of transfer matrix method, was employed. The optical
spectra were computed in the range $4.215\leq\lambda\leq 4.615\;\mu{\rm m}$.

We put emphasis on the two issues: (i) optimizing the peak $A$ at
$\lambda_{0}=4.415\;\mu$m and $\lambda_{0}=4.500\;\mu$m, which are kept fixed,
while fulfilling Eq.(1); (ii) seeking the designs with a high tolerance of the
efficiency to errors in the grating fabrication. Interim design parameters
serving as trial ones for computer aided optimization, will be obtained using
the reflection phases [see Eq.(1)] of stand-alone mirrors unlu , which means
the mirrors put on an infinite medium of the same material as the cavity’s
material (CdTe in our designs) and irradiated from the substrate.

##  3 Conventional RCE PD structures

The reported RCE HgCdTe-PDs faraone1 ; faraone2 ; faraone3  have a standard
configuration in which the CdTe FP cavity embeds a thin Hg0.71Cd0.29Te layer
between two DBR mirrors [see in Fig.1(a)]. The HgCdTe/CdTe bilayers provide
good lattice matching but the small difference in RIs of Hg0.56Cd0.44Te and
CdTe requires $m=30$, i.e. $\sim 20\;\mu$m thick DBR mirror, in order to
achieve the desired reflectivity faraone3 . Apart from enlarging the cavity,
the deposition of such a thick multilayer inevitably produces growth defects
which deteriorate the reflectance. In practice, such DBRs have been fabricated
with $m=10$ kanev  and $m=7$ faraone3  and optimized with $m=8$ for high
temperature-operation RCE PDs sioma .

Figure 1: RCE HgCdTe-absorber structures with the: (a) (HL)kH DBR and (b)
grating front mirror; the back mirror in both is a (HL)mH DBR; the irradiation
is from a CdZnTe substrate.

We adopted a mid value of $m=15$, while due to faraone1 ; faraone2 ; faraone3
$k=2$, $t_{\rm a}=75\;{\rm nm}$. For a standard placement of the absorbing
layer in the middle of the cavity ($d_{\rm f}=d_{\rm b}$), combining
semianalytical calculation and manual optimization, we found the optimal
cavity length for which $A=81.9\%$ at $\lambda_{0}=4.415\;\mu\rm m$ [RCE-S
structure in Table 2]. The automatic optimization results in the same $t_{\rm
c}$ as obtained manually. Further optimization, allowing for $d_{\rm f}\neq
d_{\rm b}$, yields RCE-Oa and RCE-Ob structures with $\lambda_{0}=4.415$ and
$\lambda_{0}=4.500\;\mu\rm m$, respectively [see in Fig.3 and Table 2]. Figure
3 shows the $A$ spectra of the designed structures along with the reflectance
($R$) from the stand-alone front DBR mirror.

Table 1: The refractive indexes $N=n+ik$ used in the simulations Wavelengths | Ge | SiO | CdTe | Hg0.71Cd0.29Te | Hg0.56Cd0.44Te | CdZeTe
---|---|---|---|---|---|---
$4.415\;\mu\rm m$ | $3.9332$ | $1.78$ | $2.6695$ | $3.4826+i0.1477$ | $2.9709$ | 2.6896
$4.500\;\mu\rm m$ | $3.9325$ | $1.78$ | $2.6691$ | $3.4665+i\;0.1425$ | $2.9697$ | 2.6895

##  4 Grating mirror based RCE PD structures

Figure 1(b) presents a modified RCE HgCdTe-PD, in which the front mirror is
replaced by a one-dimensional (1-D) dielectric grating with period $\Lambda$,
groove width $W$ and depth $t_{\rm g}$, while the absorber, back mirror and
irradiation scheme remain the same as those of the structure in Sec. 3. The
grating is designed as the stand-alone $\sim 100$ % reflectivity mirror. Our
design of the RCE absorber as a whole is done for the backside irradiation
from CdZnTe substrate and is not suitable for the front irradiation from air
[see Fig.1], because the grating mirror breaks down the irradiation symmetry
inherent to the DBR based FP cavities.

###  4.1 Polarization-selective RCE HgCdTe-absorber structures

Earlier work havaus  and a recent paper chang-a  on designing grating mirrors
have considered optical incidence from air, but here the stand-alone grating
mirror is irradiated from CdTe. To get rid of the grating reflection orders,
we imposed over all the computed wavelength range the subwavelength grating
restriction $n_{\rm CdTe}\Lambda<\lambda$, which is tighter than for the
incidence from air. Because of polarization sensitivity of the 1-D gratings,
we first maximized the $R$ for the incidence from CdTe half-space, separately
for the TE and TM polarization.

Figure 2: The spectra of reflectance from the front DBR mirrors and of
absorbance for the RCE-Oa and RCE-Ob structures.

Figure 3: The spectra of reflectance from the front grating mirrors and of
absorbance for the RCE-TEa and RCE-TEb structures.

Other trial dimensions were estimated semianalytically using the $\varphi_{\rm
f}$ and $\varphi_{\rm b}$ as computed numerically for the stand-alone grating
and DBR, respectively, and then refined by the optimization. The manual design
was performed under the constraints: $d_{\rm f}=d_{\rm b}$, $w=W/\Lambda=0.5$
and $t_{\rm a}=0.075\;\mu{\rm m}$, by controlling the $t_{\rm c}$ which
allowed us to attain the desired $\delta_{0}$. To remind, the optimization
goal is to achieve maximal polarized $A(\lambda_{0})$, minimal changes of
which due to variations in the grating fabrication process are being very
desirable. A fully computer-aided optimization procedure easily allows for
designing the structures with $d_{\rm f}\neq d_{\rm b}$ and $w\neq 0.5$. The
results of such an optimization are the structures RCE-TEa and RCE-TMa
($\lambda_{0}=4.415\;\mu$m), RCE-TEb and RCE-TMb ($\lambda_{0}=4.500\;\mu$m)
for operation with a TE and TM polarized radiation [see Table 2].

The TE and TM polarized $R$ spectra from the stand-alone grating mirrors and
the related $A$ spectra of the RCE-TEa and RCE-TEb structures are shown in
Fig. 3 and 5, respectively. The designed gratings maintain the polarized
$R\geq 99\%$ in a wide band around $\lambda_{0}$. For the TM polarization, as
seen in Fig. 5, the high-$R$ band width greatly exceeds $0.3\;\mu$m, in a good
agreement with the paper chang-a . For the TE polarization, however, this band
is narrower which does not prevent from achieving the $A\approx 100$% peak
[see Fig. 3] and a superior tolerance to a variation in the grating
fabrication process, as discussed in Subsect. 4.2 below.

###  4.2 Design tolerances

An important issue for the practicality of the proposed RCE HgCdTe-PDs is the
sensitivity to variations in the grating fabrication process. In this paper,
we report the tolerance of the value and position of the $A$ peak to the
grating groove duty cycle $w$ variations. The tolerance was found to be very
good for the structures with the optimal $w\neq 0.5$, especially in the TE
case. For example, the RCE-TEa structure still maintains high values of
$A(\lambda_{0})=93.8$% and $98.8$% with the initially prescribed $\lambda_{0}$
and $w=0.528$ and $w=0.728$, respectively. The small drop in $A(\lambda_{0})$,
should not be treated as a deterioration of the detection performance since
the peak $A=99$% appears to be shifted by at most 3 nm from the $\lambda_{0}$,
which indicates an excellent tolerance including a small change of the peak
value and a minute shift of the peak position. The RCE-TEb structure exhibits
similar tolerance properties.

Figure 4: The reflectance from the stand-alone grating mirrors and the
absorbance of the RCE-TMa and RCE-TMb structures.

Figure 5: The peak absorbance and wavelength versus $w$ for the RCE-TMb
structure.

The RCE-TMa,b structures withstand errors in $w$ of $\pm 0.05$ as related to
peak $A$, which indicates a lower but still practical tolerance. However, the
peak wavelength shift proves notably larger as compared to RCE-TEa,b
structures. For example, as $w$ is shifted to $0.287$ the RCE-TMa structure
exhibits a $24$ nm red shift of the absorption peak and a drop of the peak $A$
to $97.5\%$. In this case, we found numerically that the peak wavelength,
within a reasonable range, depends on $w$ linearly, as shown in Fig. 5.

##  5 Conclusion

The RCE HgCdTe-PDs, including new ones in which the front mirror is a $\sim
100\%$ reflectance grating, for applications in MWIR range, were designed and
simulated using semianalytic and computer-aided tools such as a scrutinized
control of the resonant round-trip phase $\delta_{0}$, and the grating
reflection amplitude and phase. The results show that the $\delta_{0}$ control
is crucial for optimizing the efficiency, as was known for the conventional
RCE PDs unlu ; faraone1 ; faraone2 ; faraone3 . In our case, $n_{\rm a}$ and
$n_{\rm c}$ differ significantly, but the reflections from the absorbing layer
have a relatively small effect due to $t_{\rm a}\ll t_{\rm c}$. We also proved
that a non-standard placement of the absorbing layer is capable of increasing
the efficiency of the conventional RCE HgCdTe-PDs, which still remains below
$85.2$%.

It was decisively shown that for a linearly polarized light, the grating
mirror based RCE HgCdTe-PDs can be designed to achieve $\sim 100\%$
efficiency, thus highly outperforming the conventional RCE HgCdTe-PDs, and
simultaneously is highly tolerant to variations in the grating groove duty
cycle $w$. For the TM designs, a linear dependence was found between the $w$
varied around its optimal value and the peak wavelength shifting, in a
response, around the prescribed $\lambda_{0}$.

Table 2: The designed and simulated RCE HgCdTe-absorber structures. The dimensional parameters and $\lambda_{\rm{R}}$ are in microns. lowercase a and b refers to a wavelengths of $4.415\;\mu$m and $4.500\;\mu$m respectively. Structures | $d_{\rm{b}}$ | $t_{\rm{a}}$ | $d_{\rm{f}}$ | $t_{\rm{Ge}}$ | $t_{\rm{g}}$ | $\Lambda$ | $w$ | Peak $A$ | $\lambda_{\rm{R}}$
---|---|---|---|---|---|---|---|---|---
RCE-S | 1.177 | 0.075 | 1.177 | - | - | - | - | 0.819 | -
RCE-Oa | 0.272 | 0.075 | 0.433 | - | - | - | - | 0.831 | -
RCE-Ob | 0.292 | 0.075 | 0.429 | - | - | - | - | 0.828 | -
RCE-TEa | 0.209 | 0.075 | 0.535 | 0.331 | 0.844 | 1.461 | 0.628 | 0.998 | 3.901
RCE-TEb | 0.228 | 0.075 | 0.534 | 0.335 | 0.861 | 1.492 | 0.629 | 0.999 | 3.984
RCE-TMa | 0.259 | 0.075 | 0.504 | 0.628 | 0.921 | 1.456 | 0.337 | 0.999 | 3.888
RCE-TMb | 0.286 | 0.075 | 0.495 | 0.650 | 0.930 | 1.468 | 0.334 | 0.999 | 3.920

## References

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  * (9) M. Auslender and S. Hava, Scattering-matrix propagation algorithm in full-vectorial optics of multilayer grating structures, Opt. Lett., 21, 1765-1767 (1996).
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# Sharp asymptotics for the free energy of 1+1 dimensional directed polymers
in an infinitely divisible environment

###### Abstract

We give sharp estimate for the free energy of directed polymers in random
environment in dimension 1+1. This estimate was known for a Gaussian
environment, we extend it to the case where the law of the environment is
infinitely divisible.

Key words. directed polymers in random environment, free energy, infinite
divisibility, FKG inequality

2000 AMS Subject Classifications. 60K37, 82D30, 60E07

##  1 Introduction

We refer to [6] for a review of directed polymers in random environment. Let
$S=(S_{n})_{n\in\mathbb{N}}$ be the simple random walk on $\mathbb{Z}^{d}$
starting at $0$, defined on the probability space $(\Sigma,\cal
E,\textbf{P})$. Let
$\eta=(\eta(n,x))_{(n,x)\in\mathbb{N}\times\mathbb{Z}^{d}}$ be a sequence of
real-valued i.i.d. random variables defined on another probability space
$(\Omega,\cal F,\mathbb{P})$. The expectation of a function $f$ with respect
to the probability measure P (respectively $\mathbb{P}$) will be denoted by
$\textbf{E}f=\int_{\Sigma}fd\textbf{P}$ (respectively
$\mathbb{E}f=\int_{\Omega}fd\mathbb{P}$). The path $(i,S_{i})_{1\leq i\leq n}$
represents the directed polymer of size $n$ in dimension $1+d$, and $\eta$ is
the random environment. For $\beta$ strictly positive, featuring the inverse
of the temperature, we define the random polymer probability measure
$\mathbf{P}_{n,\beta}$ on the path space $(\Sigma,\cal E)$ by its density with
respect to $\mathbf{P}$

$\frac{d\mathbf{P}_{n,\beta}}{d\textbf{P}}(S)=\displaystyle\frac{1}{Z_{n}(\beta)}\exp(\beta
H_{n}(S)),$ (1.1)

where

$H_{n}(S)=\sum_{j=1}^{n}\eta(j,S_{j}),\textrm{ and
}Z_{n}(\beta)=\textbf{E}\exp(\beta H_{n}(S)).$ (1.2)

For a given realisation of the environment $\eta$, the measure
$\mathbf{P}_{n,\beta}$ gives heavier weight to polymer paths $(i,S_{i})_{1\leq
i\leq n}$ with lower energy $-H_{n}(S)$ (configurations of lowest energy are
the most probable). For simplicity we write
$\mathbb{E}f(\eta)=\mathbb{E}f(\eta(0,0))$ for any $f$ such that
$f\circ\eta(0,0)$ is integrable. Let
$\lambda(\beta)=\ln\mathbb{E}e^{\beta\eta}$ be the logarithmic moment
generating function of $\eta$. We suppose that there exists $B>0$ such that

$\mathbb{E}e^{B|\eta|}<\infty\textrm{ for }0\leq\beta\leq B.$ (1.3)

It is well known that the sequence $\mathbb{E}\ln Z_{n}(\beta)$ is
superadditive, hence the limit

$p(\beta)=\lim\limits_{n\rightarrow\infty}\frac{1}{n}\mathbb{E}\ln(Z_{n}(\beta))=\sup_{n}\frac{1}{n}\mathbb{E}\ln(Z_{n}(\beta))\in(-\infty,\lambda(\beta)]$
(1.4)

exists. It is called the free energy of the polymer. We proved in [15], [18]
that if $\mathbb{E}e^{\beta|\eta|}<\infty$ for a fixed $\beta>0$, then there
exists $K>0$ such that for all $n\geq 1$,

$\mathbb{P}(\pm\frac{1}{n}(\ln Z_{n}(\beta)-\mathbb{E}\ln
Z_{n}(\beta))>x)\leq\left\\{\begin{aligned}
&\exp(-\displaystyle\frac{nx^{2}}{4K})&\textrm{ if }&x\in(0,2K],\\\
&\exp(-n(x-K))&\textrm{ if }&x\in(2K,\infty).\end{aligned}\right.$

This concentration property implies that under our assumption, $\frac{1}{n}\ln
Z_{n}(\beta)$ converges $\mathbb{P}$ a.s. towards $p(\beta)$ for every
$\beta\in(0,B)$. This was first proved by Carmona and Hu ([4], Proposition
1.4) for a Gaussian environment, and by Comets, Shiga and Yoshida ([5],
Proposition 2.5) for a general environment, but under the condition that
$\mathbb{E}e^{3\beta|\eta|}<\infty$.
We consider the normalized partition function

$W_{n}(\beta)=\frac{Z_{n}(\beta)}{\mathbb{E}Z_{n}(\beta)}=Z_{n}(\beta)e^{-n\lambda(\beta)},$

and the normalized free energy

$p_{-}(\beta)=\lim\limits_{n\rightarrow\infty}\frac{1}{n}\mathbb{E}\ln(W_{n}(\beta))=p(\beta)-\lambda(\beta)\in(-\infty,0].$

Bolthausen [2] noticed that $(W_{n})$ is a positive martingale, hence
converges $\mathbb{P}$ a.s. towards a variable $W_{\infty}$, and that moreover
$\mathbb{P}(W_{\infty}=0)$ is $0$ or $1$. When there is no random environment,
i.e. when $\beta=0$, the normalized partition function $W_{n}$ is constantly
equal to one. Accordingly we say that weak disorder holds when
$\mathbb{P}(W_{\infty}=0)=0$, strong disorder holds when
$\mathbb{P}(W_{\infty}=0)=1$. It is immediate that if weak disorder holds,
then the normalized free energy $p_{-}(\beta)$ equals zero. Comets and Yoshida
proved monotonicity in $\beta$ concerning both the dichotomy between weak and
strong disorder, and the normalized free energy.

Theorem ([9], Theorem 3.2)

  1. 1.

There exists a critical value $\beta_{0}=\beta_{c}(d)\in[0,\infty]$ with
$\beta_{0}=0$ for $d=1,2$, $0<\beta_{0}\leq\infty$ for $d\geq 3$, such that
$\mathbb{P}(W_{\infty}=0)=0$ if $\beta\in\\{0\\}\cup(0,\beta_{0})$,
$\mathbb{P}(W_{\infty}=0)=1$ if $\beta>\beta_{0}$.

  2. 2.

The normalized free energy $p_{-}(\beta)$ is non-increasing in
$\beta\in[0,\infty)$. In particular there exists $\beta_{c}$ with
$\beta_{0}\leq\beta_{c}\leq\infty$ such that $p_{-}(\beta)=0$ if
$0\leq\beta\leq\beta_{c}$, $p_{-}(\beta)<0$ if $\beta>\beta_{c}$.

It is widely believed that the two critical values $\beta_{0}$ and $\beta_{c}$
are equal, but it is still an open question in dimension $d$ greater than or
equal to three. The equality $\beta_{0}=\beta_{c}=0$ is true in dimension one
and two. This was proved by Comets and Vargas ([7]) for the dimension one, and
by Lacoin ([13]) for the dimension two. Moreover in this paper Lacoin improves
the result of Comets and Vargas by giving sharp estimates of the normalized
free energy at high temperature (that is for small $\beta$). He proves the
following theorem.

Theorem ([13], Theorem 1.4 and Theorem 1.5)

  1. 1.

When $d=1$ and (1.3) holds, there exist constants $c$ and $B_{0}<B$ such that
for every $\beta$ in $[0,B_{0})$,

$-\frac{1}{c}\beta^{4}[1+(\ln\beta)^{2}]\leq p_{-}(\beta)\leq-c\beta^{4}.$
(1.5)

  2. 2.

When $d=1$ and the environment is Gaussian, then there exists a constant $c$
such that for every $\beta\leq 1$,

$-\frac{1}{c}\beta^{4}\leq p_{-}(\beta)\leq-c\beta^{4}.$ (1.6)

Our aim is to get rid of the logarithmic factor of the lower bound in the case
of a general environment. To prove the lower bound in (1.5) Lacoin combines
the second moment method and a directed percolation argument, whereas in (1.6)
he uses a specific Gaussian approach similar to what is done in [17]. More
precisely his proof relies on two inequalities, both obtained using Gaussian
integration by parts. We are able to generalize these inequalities in the case
of an infinitely divisible environment. The first inequality is still obtained
by an integration by parts formula, valid for infinitely divisible
distributions, that we learned in [3]. Actually, infinite divisibility is not
required for the second inequality, which we prove by using the FKG
inequality, in the manner of [9].

###### Theorem 1.1

When $d=1$ and $\eta$ has an infinitely divisible distribution satisfying
(1.3), there exist constants $C>0$ and $B_{0}<B$ such that for every $\beta$
in $(0,B_{0})$,

$-C\beta^{4}\leq p_{-}(\beta).$ (1.7)

The idea of the proof is to adapt techniques which in the framework of spin
glasses are known as replica-coupling and interpolation (introduced in [12]).
Toninelli adapted these techniques to prove disorder irrelevance for the
disordered pinning model in the case where the disorder has a Gaussian
distribution ([17], Theorem 2.1), and mentioned that he can extend his result
to the situation where the environment is bounded. We notice that we can hope
to adapt his proof of the Gaussian case to the case where the environment is
infinitely divisible, in the same manner as we adapted the proof of Lacoin. We
underline the fact that replica-coupling is not a purely Gaussian technique,
and can be adapted to the more general setting of infinite divisible
environments. This was already a key ingredient in [3]. Concerning directed
polymers, the Brownian polymer in Poisson environment ([8]) might be another
good candidate to apply the replica-coupling method. Let us finally mention
that recent work on 1+1 dimensional polymers (for instance [1] and [16]) give
another perspective on the $\beta^{4}$ scaling.

The paper is organized as follows. In the first part we explain the proof of
Theorem 1.1, reducing it to the proof of the two inequalities mentioned above.
In the second part we prove the first inequality, using the integration by
parts formula for infinitely divisible distributions (Lemma 2.1). In the third
part we prove the second inequality by using the FKG inequality (Lemma 2.2).

##  2 Proof of Theorem 1.1

We can assume without loss of generality that the variance $\sigma^{2}$ of
$\eta$ is strictly less than one. If it is not the case, we consider
$\tilde{\eta}=\frac{\eta}{2\sigma}$ whose variance is $\frac{1}{4}$ and we use
that $p_{-}(\beta)=\tilde{p}_{-}(2\sigma\beta)$. We define

$p_{n}(\beta)=\frac{1}{n}\mathbb{E}\ln W_{n}(\beta)=\frac{1}{n}\mathbb{E}\ln
Z_{n}(\beta)-\lambda(\beta)$ (2.8)

and we recall that

$p_{-}(\beta)=\sup_{n\geq 1}p_{n}(\beta).$

We use the same strategy as Lacoin in [13] and show that there exist $B_{1}>0$
and $c>0$ such that for every $n\geq 1$ and every $\beta\in[0,B_{1})$,

$p_{n}(\beta)\geq(1-e^{c})\frac{1}{2n}\ln E^{\otimes
2}e^{2\beta^{2}L_{n}(S^{1},S^{2})},$ (2.9)

where $S^{1}$, $S^{2}$ are two independent copies of the simple random walk,
$E^{\otimes 2}$ is the expectation on the product space $(\Sigma^{2},\cal
E^{\otimes 2})$, and

$L_{n}(S^{1},S^{2})=\sum_{i=1}^{n}\mathbf{1}_{S_{i}^{1}=S_{i}^{2}}$

is the number of intersections of the two directed paths $(i,S_{i}^{1})_{1\leq
i\leq n}$ and $(i,S_{i}^{2})_{1\leq i\leq n}$. Now $E^{\otimes
2}e^{2\beta^{2}L_{n}(S^{1},S^{2})}$ is the partition function of a homogeneous
pinning system of size $n$ and parameter $2\beta^{2}$ with underlying renewal
process the sets of zeros of the random walk $S^{1}-S^{2}$. We refer to [10],
section 1.2 for a survey of the homogeneous pinning model, where it is proved
(Proposition 1.1 and Remark 1.2) that

$\lim_{n\to\infty}\frac{1}{n}\ln E^{\otimes
2}e^{tL_{n}(S^{1},S^{2})}=F(t),\textrm{ with }F(t)\sim\frac{t^{2}}{2}\textrm{
when }t\to 0^{+}.$

Letting $n$ tend to infinity in (2.9) we find that

$p_{-}(\beta)\geq\frac{1-e^{c}}{2}F(2\beta^{2}).$

This implies that for any $C>e^{c}-1$, there exists $B_{0}<B$ such that for
all $0\leq\beta<B_{0}$,

$p_{-}(\beta)\geq-C\beta^{4},$

which proves the lower bound result (1.7). To prove (2.9) we proceed as in
[13] by interpolation between the two functions

$p_{n}(\beta)=\frac{1}{n}\mathbb{E}\ln Ee^{\beta
H_{n}(S)-n\lambda(\beta)}=\frac{1}{2n}\mathbb{E}\ln E^{\otimes 2}e^{\beta
H_{n}(S^{1},S^{2})-2n\lambda(\beta)},$

where

$H_{n}(S^{1},S^{2})=H_{n}(S^{1})+H_{n}(S^{2}),$

and

$F_{n}(\beta)=\frac{1}{2n}\ln E^{\otimes
2}e^{2\beta^{2}L_{n}(S^{1},S^{2})}=\frac{1}{2n}\mathbb{E}\ln E^{\otimes
2}e^{2\beta^{2}L_{n}(S^{1},S^{2})}.$

Let $\beta$ be fixed in $]0,B[$. We define, for $t\in[0,1]$ and $u\geq 0$,

$\varphi_{n}(t,u)=\frac{1}{2n}\mathbb{E}\ln E^{\otimes 2}e^{\sqrt{t}\beta
H_{n}(S^{1},S^{2})-2n\lambda(\sqrt{t}\beta)+u\beta^{2}L_{n}(S^{1},S^{2})},$
$\phi_{n}(t)=\varphi_{n}(t,0)=\frac{1}{n}\mathbb{E}\ln Ee^{\sqrt{t}\beta
H_{n}(S)-n\lambda(\sqrt{t}\beta)}=p_{n}(\sqrt{t}\beta),$

so that

$\phi_{n}(1)=\varphi_{n}(1,0)=p_{n}(\beta),\ \ \varphi_{n}(0,2)=F_{n}(\beta).$

The inequality (2.9) amounts to

$\phi_{n}(1)\geq(1-e^{c})\varphi_{n}(0,2).$ (2.10)

We shall prove that

$\forall t\in[0,1],\ \ \phi^{\prime}_{n}(t)\geq
c(\phi_{n}(t)-\varphi_{n}(0,2)),$ (2.11)

which by Gronwall’s lemma implies (2.10). We write

$\phi_{n}(t)-\varphi_{n}(0,2)=\varphi_{n}(t,0)-\varphi_{n}(t,2-t)+\varphi_{n}(t,2-t)-\varphi_{n}(0,2)$
(2.12)

and we consider separately the two differences of the right member. To prove
(2.11) it is enough to prove the two following inequalities.

$\exists c>0,\quad\forall t\in[0,1],\ \ \phi^{\prime}_{n}(t)\geq
c(\varphi_{n}(t,0)-\varphi_{n}(t,2-t)),$ (2.13) $\forall t\in[0,1],\ \ \
\varphi_{n}(t,2-t)-\varphi_{n}(0,2)\leq 0.$ (2.14)

To prove (2.13) we need infinite divisibility whereas (2.14) is true in
general. Let us first prove (2.13). The function $u\mapsto\varphi_{n}(t,u)$ is
convex non decreasing, hence for every $t\in[0,1]$,

$\varphi_{n}(t,2-t)-\varphi_{n}(t,0)\geq(2-t)\frac{\partial\varphi_{n}}{\partial
u}(t,0)\geq\frac{\partial\varphi_{n}}{\partial u}(t,0).$ (2.15)

A simple calculation shows that

$\frac{\partial\varphi_{n}}{\partial
u}(t,0)=\frac{\beta^{2}}{2n}\mathbb{E}E_{n,\sqrt{t}\beta}^{\otimes
2}L_{n}(S^{1},S^{2}),$ (2.16)

where $E_{n,\beta}$ is the expectation with respect to the probability measure
$P_{n,\beta}$, and $E_{n,\beta}^{\otimes 2}$ is the expectation with respect
to the product measure $P_{n,\beta}^{\otimes 2}$ on the product space
$(\Sigma^{2},\cal E^{\otimes 2})$. On the other hand,

$\phi_{n}^{\prime}(t)=\frac{\beta}{2\sqrt{t}}p_{n}^{\prime}(\sqrt{t}\beta).$
(2.17)

Now the quantities (2.16) and (2.17) are related through the following lemma,
whose proof we postpone to the next section.

###### Lemma 2.1

If the law of $\eta$ is infinitely divisible, then there exists $c>0$ such
that for every $\beta\in[0,\frac{B}{2}),$

$p_{n}^{\prime}(\beta)\geq-\frac{c\beta}{n}\mathbb{E}E_{n,\beta}^{\otimes
2}L_{n}(S^{1},S^{2}).$ (2.18)

We use successively (2.17), the Lemma 2.1, (2.16), and (2.15), and we obtain
that provided we take $\beta<\frac{B}{2}$, there exists $c>0$ such that for
every $t\in[0,1]$,

$\phi_{n}^{\prime}(t)\geq-\frac{c\beta^{2}}{2n}\mathbb{E}E_{n,\sqrt{t}\beta}^{\otimes
2}L_{n}(S^{1},S^{2})\geq c(\varphi_{n}(t,0)-\varphi_{n}(t,2-t)),$

which is exactly (2.13). Now we prove the second inequality (2.14) thanks to
the following lemma.

###### Lemma 2.2

If $\sigma^{2}<1$ then there exists $B_{2}>0$ such that for every fixed
$\beta$ in $(0,B_{2})$,

$\frac{\partial}{\partial u}\varphi_{n}(t,u)-\frac{\partial}{\partial
t}\varphi_{n}(t,u)\geq 0\ \ \ \textrm{ for }t\in[0,1]\textrm{ and }u\geq 0.$
(2.19)

We postpone the proof of Lemma 2.2 and we conclude the proof of the theorem.
We set $B_{1}=\min(\frac{B}{2},B_{2})$ and fix $\beta$ in $[0,B_{1})$. Then
according to Lemma 2.2

$\varphi_{n}(t,2-t)-\varphi_{n}(0,2)=\int_{0}^{t}\frac{\partial\varphi_{n}}{\partial
t}(s,2-s)-\frac{\partial\varphi_{n}}{\partial u}(s,2-s)ds\leq 0,$

which is inequality (2.14). Combining (2.12), (2.13), and (2.14), we obtain
(2.11), which ends the proof.

###### Remark 2.3

If the law of $\eta$ is standard normal, then the inequality (2.18) is an
equality with $c=1$, and the inequality (2.19) holds true. Both facts are
showned in [13] by using Gaussian integration by parts.

##  3 Proof of Lemma 2.1

According to [3], Proposition 11, there exists $c_{2}>0$ depending on $\beta$
such that

$p_{n}^{\prime}(\beta)\geq-\frac{c_{2}}{n}\mathbb{E}E_{n,\beta}^{\otimes
2}L_{n}(S^{1},S^{2}).$ (3.20)

We recall rapidly the proof of [3]. We write

$np_{n}^{\prime}(\beta)=\mathbb{E}\frac{EH_{n}(S)e^{\beta H_{n}(S)}}{Ee^{\beta
H_{n}(S)}}-n\lambda^{\prime}(\beta)=\sum_{(i,x)}\mathbb{E}\eta(i,x)f_{i,x}(\eta(i,x))-n\lambda^{\prime}(\beta).$

The distribution of $\eta$ is infinitely divisible, therefore we have a Lévy
Khinchine formula

$\lambda(\beta)=c_{0}\beta+\frac{\sigma^{2}}{2}\beta^{2}+\int(e^{\beta
u}-1-\beta u\mathbf{1}_{\left|u\right|\leq 1})\pi(du),$

where $c_{0}\in\mathbb{R}$, $\sigma\geq 0$, and $\pi$ is a measure on
$\mathbb{R}^{*}$ such that $1\wedge u^{2}\in L^{1}(\pi)$. With these notations
we have the following lemma.

Lemma ([3], Lemma 10). For any bounded differentiable $f$ with bounded
derivative, one has the following integration by parts formula:

$\mathbb{E}\eta
f(\eta)=c_{0}\mathbb{E}f(\eta)+\sigma^{2}\mathbb{E}f^{\prime}(\eta)+\int_{\mathbb{R}}(\mathbb{E}f(\eta+u)-\mathbf{1}_{\left|u\right|\leq
1}\mathbb{E}f(\eta))u\pi(du).$

Applying the lemma, Carmona and al. calculate

$np_{n}^{\prime}(\beta)=-\sigma^{2}\beta\mathbb{E}E_{n,\beta}^{\otimes
2}L_{n}(S^{1},S^{2})-\sum_{(i,x)}\mathbb{E}(P_{n,\beta}(S_{i}=x))^{2}\int_{\mathbb{R}}\frac{(e^{\beta
u}-1)e^{\beta u}u}{(e^{\beta u}-1)\mu_{n}(S_{i}=x)+1}\pi(du).$

Noting that

$\sum_{(i,x)}(P_{n,\beta}(S_{i}=x))^{2}=E_{n,\beta}^{\otimes
2}L_{n}(S^{1},S^{2}),$

and that

$\sup_{0\leq a\leq 1}\int_{\mathbb{R}}\frac{(e^{\beta u}-1)e^{\beta
u}u}{(e^{\beta u}-1)a+1}\pi(du)\\\
\leq\int_{-\infty}^{0}\left|u\right|(1-e^{\beta
u})\pi(du)+\int_{0}^{+\infty}ue^{\beta u}(e^{\beta
u}-1)\pi(du)=c^{\prime}(\beta),$ (3.21)

which is finite provided that $\mathbb{E}e^{2\beta\eta}$ is finite, they
conclude that (3.20) is true with $c_{2}=c^{\prime}(\beta)+\sigma^{2}\beta$.

Now for $u<0$ we have $1-e^{\beta u}\leq\beta\left|u\right|$, and for $u>0$ we
have $e^{\beta u}-1\leq\beta ue^{\beta u},$ so returning to (3.21) we deduce
that for every $\beta\in[0,\frac{B}{2}]$,

$c^{\prime}(\beta)\leq\beta\left(\int_{-\infty}^{0}u^{2}\pi(du)+\int_{0}^{+\infty}u^{2}e^{Bu}\pi(du)\right),$

and conclude that (2.18) is satisfied with

$c=\sigma^{2}+\int_{-\infty}^{0}u^{2}\pi(du)+\int_{0}^{+\infty}u^{2}e^{Bu}\pi(du).$

##  4 Proof of Lemma 2.2

We calculate

$\frac{\partial}{\partial u}\varphi_{n}(t,u)-\frac{\partial}{\partial
t}\varphi_{n}(t,u)=\frac{1}{2n}E^{\otimes
2}[e^{u\beta^{2}L_{n}(S^{1},S^{2})}\mathbb{I}(S^{1},S^{2})],$ (4.22)

where

$\displaystyle\mathbb{I}(S^{1},S^{2})$
$\displaystyle=\mathbb{E}\frac{e^{\sqrt{t}\beta
H_{n}(S^{1},S^{2})-2n\lambda(\sqrt{t}\beta)}(\beta^{2}L_{n}(S^{1},S^{2})-\frac{\beta}{2\sqrt{t}}H_{n}(S^{1},S^{2})+\frac{\beta
n}{\sqrt{t}}\lambda^{\prime}(\sqrt{t}\beta))}{E^{\otimes 2}e^{\sqrt{t}\beta
H_{n}(S^{1},S^{2})-2n\lambda(\sqrt{t}\beta)+u\beta^{2}L_{n}(S^{1},S^{2})}}$
$\displaystyle=\mathbb{E}\frac{e^{\sqrt{t}\beta
H_{n}(S^{1},S^{2})}(\beta^{2}L_{n}(S^{1},S^{2})-\frac{\beta}{2\sqrt{t}}H_{n}(S^{1},S^{2})+\frac{\beta
n}{\sqrt{t}}\lambda^{\prime}(\sqrt{t}\beta))}{E^{\otimes 2}e^{\sqrt{t}\beta
H_{n}(S^{1},S^{2})+u\beta^{2}L_{n}(S^{1},S^{2})}}$

We show that $\mathbb{I}(S^{1},S^{2})$ is non negative for every
$(S^{1},S^{2})$ in $\Sigma^{2}$. Let $(S^{1},S^{2})$ be fixed in $\Sigma^{2}$.
We define

$I=\\{i\in[\\![1,n]\\!];S^{1}_{i}=S^{2}_{i}\\}\textrm{ and }A=\\{(i,x);1\leq
i\leq n\textrm{ and }x=S^{1}_{i}\textrm{ or }S^{2}_{i}\\}.$

We write $H_{n}$ for $H_{n}(S^{1},S^{2})$ and $L_{n}$ for
$L_{n}(S^{1},S^{2})$. For $v=(v_{(i,x)})_{(i,x)\in A}$, we define

$H_{n}(v)=\sum_{i=1}^{n}v_{(i,S^{1}_{i})}+v_{(i,S^{2}_{i})}=\sum_{i\in
I}2v_{(i,S^{1}_{i})}+\sum_{i\notin I}v_{(i,S^{1}_{i})}+v_{(i,S^{2}_{i})}$

Then the measure defined by

$\mu(dv)=\frac{e^{\sqrt{t}\beta H_{n}(v)}}{\mathbb{E}(e^{\sqrt{t}\beta
H_{n}})}\mathbb{P}_{\eta}^{\otimes(2n-L_{n})}$

is a product probability measure on $\mathbb{R}^{2n-L_{n}}$, and we can write
$\mathbb{I}(S^{1},S^{2})$ as an expectation with respect to $\mu$:

$\mathbb{I}(S^{1},S^{2})=\int\frac{\beta^{2}L_{n}-\frac{\beta}{2\sqrt{t}}H_{n}(v)+\frac{\beta
n}{\sqrt{t}}\lambda^{\prime}(\sqrt{t}\beta)}{E^{\otimes 2}e^{\sqrt{t}\beta
H_{n}(v)+u\beta^{2}L_{n}}}\mathbb{E}(e^{\sqrt{t}\beta H_{n}})\mu(dv)\\\ =\int
X(v)Y(v)\mu(dv),$ (4.23)

with

$X(v)=\frac{1}{E^{\otimes 2}e^{\sqrt{t}\beta H_{n}(v)+u\beta^{2}L_{n}}}$

and

$Y(v)=\Big{(}\beta^{2}L_{n}-\frac{\beta}{2\sqrt{t}}H_{n}(v)+\frac{\beta
n}{\sqrt{t}}\lambda^{\prime}(\sqrt{t}\beta)\Big{)}\mathbb{E}(e^{\sqrt{t}\beta
H_{n}}).$

As $X$ and $Y$ are decreasing random variables in $L^{2}(\mu)$, we deduce from
the FKG inequality that

$\int X(v)Y(v)\mu(dv)\geq\int X(v)\mu(dv)\times\int Y(v)\mu(dv).$ (4.24)

We refer to [11], Theorem 2.4 or [14], chapter II for more information about
the FKG inequality. We calculate $\int Y(v)\mu(dv)$ and find that

$\int Y(v)\mu(dv)=\Big{(}\beta^{2}L_{n}+\frac{\beta
n}{\sqrt{t}}\lambda^{\prime}(\sqrt{t}\beta)\Big{)}\mathbb{E}(e^{\sqrt{t}\beta
H_{n}})-\frac{\beta}{2\sqrt{t}}\mathbb{E}(H_{n}e^{\sqrt{t}\beta H_{n}}).$

Using that $\mathbb{E}(\eta e^{u\eta})=\lambda^{\prime}(u)e^{\lambda(u)}$, we
calculate

$\mathbb{E}(H_{n}e^{\sqrt{t}\beta
H_{n}})=\Big{(}2L_{n}\lambda^{\prime}(2\sqrt{t}\beta)+2(n-L_{n})\lambda^{\prime}(\sqrt{t}\beta)\Big{)}\mathbb{E}(e^{\sqrt{t}\beta
H_{n}}),$

hence

$\int
Y(v)\mu(dv)=\beta^{2}L_{n}\Big{(}1-\frac{\lambda^{\prime}(2\sqrt{t}\beta)-\lambda^{\prime}(\sqrt{t}\beta)}{\sqrt{t}\beta}\Big{)}\mathbb{E}e^{\sqrt{t}\beta
H_{n}}.$

There exists $a\in(\sqrt{t}\beta,2\sqrt{t}\beta)$ such that
$\frac{\lambda^{\prime}(2\sqrt{t}\beta)-\lambda^{\prime}(\sqrt{t}\beta)}{\sqrt{t}\beta}=\lambda^{\prime\prime}(a)$.
Since $\lambda^{\prime\prime}(0)=\sigma^{2}$ is strictly less than one there
exists $B_{2}>0$ such that for every $s$ in $[0,2B_{2})$,
$\lambda^{\prime\prime}(s)$ remains strictly less than one. If
$\beta\in(0,B_{2})$ then $a\in(0,2B_{2})$, hence
$\lambda^{\prime\prime}(a)<1$, from which we deduce that $\int Y(v)\mu(dv)$ is
non negative. As $\int X(v)\mu(dv)$ is also obviously non negative, recalling
(4.22), (4.23) and (4.24), we get the result.

## Acknowledgements

The author thanks very much the anonymous referee for his careful reading and
valuable comments.

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