Cristian Olariu
email: cristian.olariu@ucd.ie

Adriana Hava
email: adriana.hava@ucdconnect.ie

Piotr Szczechowiak

Yacine Ghamri-Doudane

Prototyping Telematic Services in a Wireless Vehicular Mesh Network Environment

Next generation telematic services are expected to play a key role in future automotive applications. In order to achieve strong integration between the services and the underlying network infrastructure there is a need for both simulation and emulation of the entire system. This paper presents a combined simulation and emulation approach for telematic services prototyping in an emulated wireless vehicular mesh networking environment. The ns-3 wireless mesh model, SUMO vehicular mobility model and different telematic services are integrated to demonstrate high scalability and flexibility of the proposed approach.

I. INTRODUCTION

With the advances in car automation there is a growing demand for networked telematic services to facilitate navigation, improve car safety and provide infotainment data to mobile users. The dynamic nature of vehicular networks creates also an opportunity to provide location-based services. These nextgeneration telematic services are expected to be an integral part of tomorrows end-user communication patterns, ensuring continuity of services usually available to end-users in home and other public places.

The current offer of telematic services for automotive users is very limited, but it is expected to expand significantly in the near future. Users driving around metropolitan areas will be able to receive safety alerts and traffic updates from local authorities while downloading infotainment data (e.g. news, music, video) from local and remote servers. The information contained in the announcements will differ based on the location of users and their preferences. This vision of telematic services offering over a wireless mesh network is presented in Figure 1.

An essential point that would contribute to the success of telematic services in such environment lies in a strong service integration and good coupling between the services and the underlying network infrastructure. This integration can be achieved by appropriate design of the services and extensive simulation and emulation of the entire system. In this paper we present the evaluation results of a system, which enables an integrated telematic services offering in a metropolitan area using a wireless mesh network as the backhaul. The main goal Figure 1. The vision of telematic services offering over a wireless mesh network of this work is to design a joint emulation/simulation framework, which will serve as a platform for prototyping different service distribution protocols, mobility support mechanisms and QoS solutions. The emulation aspect of the framework is the most important element of the system, which will enable testing in close to real conditions.

Current wireless mesh networking technologies can offer high bandwidth communication links for mobile users without requiring many interconnections into a wired infrastructure. The deployments of such networks can be also seen as a low cost alternative to cellular systems especially in dense urban environments. Mesh networks can be easily deployed by different stake holders (local authorities, network operators, ISPs etc.) to deliver not only Internet access but also other services such as safety announcements and traffic updates. One of the main advantages of wireless mesh networks is the ability to manage the services in a centralized or distributed manner, depending on the nature of the application. This gives a great flexibility in designing the service architecture and the mesh network topology. The joint emulation/simulation framework presented in this paper will serve as the main tool in the design and prototyping process of such a system.

The rest of the paper is organized as follows: Section II presents the related work. Section III describes the general approach to joint emulation/simulation of distributed systems. Section IV discusses our emulation setup. In Section V we present the evaluation results of three telematic services tested in our system. Finally we conclude in Section VI with a brief description of future work.

II. RELATED WORK

The majority of research work in wireless networking is performed through simulations. Emulation is not that frequently used due to larger complexity and a difficult development environment. In the literature we can find papers which investigate the accuracy of different simulation/emulation programs.

The work presented in [START_REF] Ivanov | Experimental validation of the ns-2 wireless model using simulation, emulation, and real network[END_REF] is an experimental evaluation of the wireless model implemented in ns-2, which compares the results of a real system, an emulated network and a simulator. It is proven that a good parametrization of the simulation model can give a good approximation of a real network behaviour. Emulation takes also into consideration the delays introduced by the operating system and the hardware itself. Therefore system emulation will provide more accurate results than simulation. In our work the wireless mesh model is implemented using the ns-3 simulator while the mobile clients and their servers are emulated using real computers.

Judd et al. [START_REF] Judd | Repeatable and realistic wireless experimentation through physical emulation[END_REF] proposed a hardware-based emulator, which mimics the characteristics of wireless links and assures realistic and repeatable experiments. The framework described in [START_REF] Judd | Repeatable and realistic wireless experimentation through physical emulation[END_REF] provides real-time modelling of the wireless communication channel between all pairs of nodes that are attached to the emulator. In order to reduce the effects of the environment the authors use cables attached to the antenna port of the wireless cards instead of wireless links.

The work described in [START_REF]Using emulation to understand and improve wireless networks and applications[END_REF] extends the study presented in [START_REF] Judd | Repeatable and realistic wireless experimentation through physical emulation[END_REF]. It enables the analysis of higher layers performance in real networks and facilitates the development and evaluation of different wireless networking protocols.

Zhou et al. in [START_REF] Zhou | Twine: A hybrid emulation testbed for wireless networks and applications[END_REF] present an efficient emulation framework called TWINE, which combines emulation with repeatability of simulations and physical components, in order to achieve scalability for wireless network systems. It emulates the wireless channel, the MAC layer and the physical layer but only supports the 802.11b WLAN standard, which limits its applicability. The three papers presented above focus mainly on designing an emulator which imitates wireless links characteristics. In our emulation setup the main goal is on prototyping different telematic services and performance evaluation of the wireless mesh and the networking protocols working on top of it.

MobiNet, presented in [START_REF] Mahadevan | Mobinet: a scalable emulation infrastructure for ad hoc and wireless networks[END_REF], is a scalable emulation infrastructure for wireless networks. It consists of core nodes, used to emulate hop-by-hop network characteristics, and edge nodes, which generate the network load and forward the packets to the core nodes. The presented framework enables mobile, ad-hoc and wireless emulation and allows researchers to experiment with a variety of MAC protocols and routing protocols, which are not usually available in a test-bed environment.

In [START_REF] Staub | Virtualmesh: an emulation framework for wireless mesh networks in omnet++[END_REF] Staub et al. presents VirtualMesh, which is an emulation for wireless mesh networks using the OMNeT++ simulator. It combines computer simulation, network emulation and virtualization into a single platform for wireless mesh network experiments. This framework is able to virtualize an entire test-bed by emulating the wireless network and running the wireless nodes as virtual hosts. Real network traffic is captured at given nodes using a virtual interface and then redirected to the OMNet++ simulator. Staub's et al. work is closely related to ours but their analysis does not show how the system behaves in the presence of mobile nodes. Our work investigates how node mobility influences protocols performance and the entire network behaviour.

The joint simulation/emulation framework presented in this paper relies on the state-of-the-art implementation performed in the ns-3 simulator. This approach allows us to use advanced functionality, which is not available in the emulation systems, which are based on OMNet++ or the ns-2 simulator. In addition, our system can simulate different mobility patterns, which can realistically emulate car movements in metropolitan areas. The presented framework is a perfect simulation/emulation tool for prototyping different telematic services and network protocols in a wireless network environment.

III. SYSTEM ARCHITECTURE

The concept of emulation can be described as a process, in which any computer platform or a program mimics the behaviour of a different computer platform or a program. It enables users to test their implementations before deploying them into the real environment. In the emulation process it is possible to attach a real network device to a simulated environment. Emulation is used when a part of the system is not available and needs to be imitated in terms of functionality. Software emulation is the most common approach, which provides emulation of software or hardware parts of a system.

Emulation is very important especially when performing research studies in the area of wireless networks. The importance of emulation in computer networks has been recently highlighted through the addition of emulation capabilities to the well known network simulators such as ns-2, ns-3 and Qualnet.

The emulated components of a computer network have to provide the same functionality as their real counterparts. The interfaces between different modules must be implemented in an appropriate way, to allow seamless operation. The emulation needs to be done in such a way that any real component will not be able to notice the differences in functionality.

Figure 2 presents the general concept of a combined simulation/emulation approach for distributed computing systems. The part of the network which does not have any connection point with the real components, is called the simulation core. A typical emulated component is the Network Interface Card (NIC) installed on a particular node. We note that usually the network device is emulated, i.e. a software component that directly links with a real network device of the simulation host (real computer hosting the simulation). The rest of the network protocol stack is provided by the simulation environment. In this manner the simulated node becomes a virtual host.

The interaction between the simulation environment and the machine, which hosts the simulation (simulation host) is done through sockets. The simulation host connects these sockets to the actual networking devices. The network devices on the simulation host grant entry points into the simulation for real hosts. This enables real traffic to flow between real hosts while being backhauled through a complex simulated topology.

Typically the simulation host needs to provide as many real network devices as emulated devices are in the simulation. In this way, real packets from real hosts can traverse the simulation core and reach nodes inside the simulation core or other real hosts. Thus, a large part of the network can be simulated while some real hosts can be virtualized.

The necessity to provide a real network device for each emulated node could become a challenge in case of large systems. However many interfaces can be set in a promiscuous mode, in which packets are not interpreted by the network stack of the real host but are directly forwarded to the sockets. In this way, one real interface can be shared by multiple emulated nodes which will be directly connected to the sockets. The combined traffic streams from the emulated nodes can be split into individual streams by a network splitter. For packet switched networks, a network switch can play the role of a splitter. One problem that might appear in such a system is when the combined data streams of emulated nodes reach the data rate limit of the real network interface card. This problem needs to be resolved at the design phase of the emulation system.

IV. FRAMEWORK DESCRIPTION

This section presents the main components of the emulation setup. The first part describes the main telematic services, which will be evaluated through emulation. The second part describes the simulation environment used in our research experiments. The last part presents the architecture of the emulated system.

A. Telematic Services description

Future telematic services for automotive users aim at improving safety on the roads through proactive, interactive, and regular announcements. They ease the access to points of interest by combining location-based services with the navigation system. They can also provide interfaces for local businesses to announce special events and offers to passing by cars. Dedicated services might be deployed to improve the driving experience around metropolitan areas through constant location-based traffic updates. On top of the above mentioned services entertainment and infotainment services can be delivered to the on-board unit in each car. In general we can classify telematic services into three broad groups:

• Safety applications; • Location Based Services;

• Infotainment services. The first group of applications is very important, because it delivers safety related information to drivers (road conditions, accidents etc.). These announcements are usually not that frequent but require guaranteed data delivery. This class of services might include also emergency VoIP calls in case of critical situations. The second group includes short periodical announcements and advertising information based on users location and their preferences. Traffic update information are also broadcasted depending on the location of the users. The last group of services has lower priority than the previous applications, but delivers popular services like music and video streaming. In order to evaluate the performance of telematic services over a wireless mesh network we decided to test one application from each group: emergency VoIP calls, video streaming, and location-based announcements.

B. Simulation model description

In order to simulate the wireless mesh network environment we used the ns-3 simulator. This tool is a discreet-event network simulator, commonly used for research purposes. Our configuration is a grid topology consisting of nine mesh nodes separated by a distance of 100 meters. The gateway is selected as the top-left node of the grid.

In our setup, the clients are represented by cars and are spread across the mesh network. The most important characteristic is that the clients have different mobility patterns and are roaming inside the mesh network along different routes. The Nakagami fast fading loss model was used as the propagation model. This is the most popular model for urban scenarios, which are the main target in this work.

Each mesh node is equipped with two 802.11g interfaces for communication between the mesh nodes and one 802.11a interface for communication with the cars. The choice of 802.11g for the communication inside the mesh network is justified by its large communication range, which is larger than the 802.11a's one. The increased number of non-overlapping channels that are available in 802.11a protocol justifies our selection of the wireless communication standard for the carclient interface. The 802.11g channels were allocated using a spread channel policy, where two non-overlapping 20MHz frequency channels are assigned to different mesh interfaces. Two interfaces are also used to improve network capacity and reliability.

The 802.11s standard implemented in ns-3 is used as the mesh network routing protocol. This employs the HWMP (Hybrid Wireless Mesh Protocol) [START_REF] Camp | The IEEE 802.11 s extended service set mesh networking standard[END_REF] as the MAC-layer routing protocol. The 802.11s standard is an extension of 802.11 for arbitrary multi-hop topologies, where each mesh node operates as a link-layer router and cooperates with all the other mesh nodes in the process of frame forwarding. HWMP is based on the AODV protocol [START_REF] Perkins | RFC 3561-ad hoc on-demand distance vector (AODV) routing[END_REF], but it works at the MAC layer for efficient path selection. HWMP can work in two modes: proactive and reactive. In the reactive mode the path discovery starts when the source wants to transmit data to an unknown destination. In the proactive mode a single mesh node is configured as a root and if the route to a destination is unknown the data is sent to the root node, which is responsible for sending it to the destination node. In all experiments we used the proactive mode of HWMP to limit the delay in the route discovery process.

We have also developed a method which uses three different tools to generate mobility patterns for the cars: SUMO (Simulation of Urban MObility) -an open-source road traffic simulation package designed to handle large road networks, OpeenStreetMap -used to obtain the XML file containing the map information from a specified area on the map, and TraNS (Traffic and Network Simulation Environment), which links SUMO and the ns-2 network simulator. TraNS allows importing an XML file and to adjust different parameters such as number of cars, delay of cars arrival and simulation time. TraNS is capable of generating mobility traces for ns-2, which can be imported into ns-3 using the Ns2MobilityHelper class.

C. Emulation system description

This subsection describes which computing and networking equipment is needed to build an emulated Wireless Mesh Network environment. In our research experiments we used three different computers. Each computer hosted one important part of the end-to-end telematic service delivery system. One computer hosted the server-side (one or more server instances can be hosted on a single computer by using virtualization software -e.g. VirtualBox), the second computer hosted the simulated network topology used to enable emulation, and the third computer hosted user applications.

To enable connectivity between each of the three computers, we used a network switch. All four above presented components, were interconnected via Ethernet cables as depicted in Figure 3. The network interface card on the PC1 was Figure 3. Emulation Setup set to the promiscuous mode. Figure 3 presents also the IP routing tables for all the computers in the emulation setup. For example purpose, let us consider one instance of a server which supplies multiple clients with services. The server will be associated with a sub-network IP address of 192.168.7.0/24 while the clients will belong to multiple subnets defined by the IP address of 192.169.0.0/16. For the proper function of the entire emulation it is essential to setup the routing tables on each computer.

V. PERFORMANCE EVALUATION

This section gives details about the emulation/simulation scenario used to obtain the performance evaluation results by running different telematic services through a simulated wireless mesh network. Figure 4 presents the mapping between the real nodes and the virtual mesh nodes simulated inside the emulation. In our simulation scenario a mobile client roams in an area of a 3x3 mesh grid (as depicted in Figure 4). For the purpose of performance evaluation of telematic services we use the mobility model, which is already implemented in ns-3. The wireless interface of the mobile node is wireless connected to an Access Point of one of the mesh nodes. A hand-over mechanism takes care of the connectivity issues related to the users' mobility. On the mobile node, a point-to-point interface is set in emulation mode. The emulated interface is connected to the simulation host's Ethernet device and through IP layer settings it is linked to the client represented by a laptop machine (the blue dot in Figure 4).

At the top-left corner of the grid there is the mesh gateway which is connected through a wired link to the simulated node. This node has an emulated point-to-point device connected to a real machine, which hosts different server applications. Table I presents the main emulation setup parameters used in our experiments. The iperf application was used on the client and server side to generate the traffic patterns, which correspond to different telematic services. In order to model the locationbased announcements we chose a UDP application sending packets with a payload of 200 Bytes and a data-rate of 1.6 kbps. From Figure 5 it can be seen that the jitter is maintained at a very low level which is sufficient for this type of service. The few spikes in the packet loss plot are due to the mobility of the car, which triggers hand-overs. Based on these results it is clear that the location-based announcement service can be supported in this kind of a network if a low percentage of packet loss is allowed. In order to improve the quality of service there is a need to design more efficient mobility support protocols.

The emergency VoIP calls were modelled as an UDP application sending packet of 100 Bytes (payload size) with a data-rate of 40 kbps. These values were chosen as a mean of the values used by current available voice codecs (e.g. AMR, G.729, G.711, etc.).

As depicted in Figure 6, the jitter is contained in the recommended limits. The tolerance for VoIP jitter is in the range of 20-30ms [START_REF]One-way Transmission Time[END_REF]. The losses occur in short bursts but more often than the ones that appear for the announcement services. This is due to the fact that packets are sent more often for the VoIP application. These results show that even in the case of time-sensitive traffic such as VoIP, the emulated environment In order to model the third group of services we used video traffic, which was generated through UDP packets of 1500 Bytes. This value represents the Maximum Transmission Unit (MTU) over a wireless medium. As the on-board-unit display is of small resolution, we considered fair to send video packets at a medium quality data-rate of 750 kbps.

The jitter for the video traffic (presented in Figure 7) is higher than for the previous services, but still in the acceptable range. The packet losses occur from the same reasons presented for the VoIP service, that is hand-overs due to user mobility. Video streaming is more demanding in terms of occupied bandwidth than a VoIP application but offers better quality of experience in similar network conditions. Our emulation environment shows that degradation of the quality metrics are present but even in this case the delivery of the services is reliable and performs in a similar way to real conditions. In this work we have presented an emulation framework which uses the latest advancements in the state-of-the-art of network simulators. This framework enables extensive evaluation of new services and networking protocols, which is difficult to perform in real tesbeds. The framework is easy to implement and has a low deployment cost. We have also described all the steps on how to build such emulation systems.

We have tested this framework in the context of telematic services and assessed its performance for several types of applications. The emulated network topology was a wireless mesh network with vehicular mobility support. The ns-3 network simulator was used to create a wireless vehicular mesh networking environment interfacing with real computers hosting the telematic services. The services were classified into three major categories: safety applications, location based services, and infotainment services. Each type of service was modelled with iperf and validated through tests using real network traffic.

With this work we have developed a joint simulation/emulation framework, which is a perfect platform for prototyping different telematic services and testing how they influence the underlying network infrastructure. This will allow strong integration of services with the network and finetuning of different network protocols and their parameters.

In the future work we plan to test different mobility models, which will emulate realistic car movements in metropolitan areas. We will also replace the synthetic traffic generated by iperf with real traffic patterns from our telematic services. We will perform large scale simulations with large number of mobile clients and mesh node deployments, which take into account the characteristics of a metropolitan area. Finally in order to validate our results we plan to compare them with the measurements acquired from a real world testbed.

Figure 2 .

 2 Figure 2. The diagram of the network emulation concept

Figure 4 .

 4 Figure 4. Visualization of the simulation and the emulated system components

Figure 5 .Figure 6 .

 56 Figure 5. Location based announcements

Figure 7 .

 7 Figure 7. Video streaming performance in a wireless mesh network

ACKNOWLEDGEMENT

This work is supported by the European Union through the Marie Curie IAPP program under the grant agreement no. 230684: "CarMesh: Ubiquitous Wireless Mesh Networks for Next-Generation Personal Digital Automotive Services".


J Najnudel
email: joseph.najnudel@math.univ-toulouse.fr

A Nikeghbali
email: ashkan.nikeghbali@math.uzh.ch

A Rouault
email: alain.rouault@uvsq.fr

LIMIT THEOREMS FOR ORTHOGONAL POLYNOMIALS RELATED TO CIRCULAR ENSEMBLES

Keywords: 2010 Mathematics Subject Classification. 15B52, 42C05, 60F10, 60F17 Random Matrices, unitary ensemble, orthogonal polynomials, large deviation principle, invariance principle

For a natural extension of the circular unitary ensemble of order n, we study as n → ∞ the asymptotic behavior of the sequence of orthogonal polynomials with respect to the spectral measure. The last term of this sequence is the characteristic polynomial. After taking logarithm and rescaling, we obtain a process indexed by t ∈ [0, 1]. We show that it converges to a deterministic limit, and we describe the fluctuations and the large deviations.

Introduction

In this paper, we study the asymptotic behavior of a triangular array of complex random variables originating from Random Matrix Theory, and more specifically from some models of unitary ensembles. Our results can be viewed as an extension of some earlier works on the characteristic polynomial of unitary matrices. Indeed, for fixed n, our random variables consist of the sequence of monic orthogonal polynomials with respect to a random measure associated with a random unitary matrix of size n, when evaluated at the point z = 1, the last term being the characteristic polynomial of that matrix.

It was already noticed in [START_REF] Bourgade | The characteristic polynomial of a random unitary matrix: a probabilistic approach[END_REF] that the characteristic polynomial of a random unitary matrix sampled from the Haar measure on U(n), the unitary group of order n has the same law as a product of independent random variables. In a previous paper ( [START_REF] Bourgade | Circular Jacobi ensembles and deformed Verblunsky coefficients[END_REF]) we saw that this characteristic polynomial Φ n (z) (evaluated at z = 1) is actually a product of variables

Φ n (1) = n-1 j=0 (1 -γ j )
and we named the independent variables γ j the modified Verblunsky coefficients, referring to the coefficients involved in the Schur recursion for orthogonal polynomials on the unit circle (OPUC), as dubbed by Simon [START_REF] Simon | Orthogonal polynomials on the unit circle. Part 1: Classical theory[END_REF]. These variables (for j < n) have an explicit density in the open unit disc, depending on their rank j. We proved in [START_REF] Bourgade | Circular Jacobi ensembles and deformed Verblunsky coefficients[END_REF] that this description stays valid when we change the probability measure on U(n), by considering the Circular Jacobi Ensemble, that we will define below. The construction of the deformed Verblunsky coefficients uses the whole sequence {Φ k,n (z), k = 0, . . . , n} of monic orthogonal polynomials with respect to the spectral measure of the pair (U, e 1 ) where e 1 is a fixed vector. We have ([5] Prop. 2.2)

Φ k,n (1) = k-1 j=0 (1 -γ j )
It is then natural to study the triangular array {Φ k,n [START_REF] Abramowitz | Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF], k = 0, . . . , n} and in order to normalize the time for different values of n, one can study the process {Φ ⌊nt⌋,n (1), t ∈ [0, 1]}, where ⌊ ⌋ denotes the integer part, and where, by convention, Φ 0,n (1) = 1.

One can note that Φ n,n is the characteristic polynomial of U and that when U is the chosen according to the Haar measure, the sequence of random variables (Φ n,n (1)) has played a crucial role in the recent interactions between random matrix theory and number theory (see [START_REF] Keating | Random matrix theory and ζ(1/2 + it)[END_REF]). It is also worthwhile to note that in [START_REF] Killip | Eigenvalue statistics for CMV matrices: from Poisson to clock via random matrix ensembles[END_REF], Killip and Stoiciu considered also a stochastic process indexed by t = k/n and related to the sequence of orthogonal polynomials in the CβE ensemble. In fact they considered variables which are the complex conjugate of our deformed Verblunsky coefficients as auxiliary variables in the study of the Prüfer phase (Lemma 2.1 in [START_REF] Killip | Eigenvalue statistics for CMV matrices: from Poisson to clock via random matrix ensembles[END_REF]). This tool is used again in [START_REF] Killip | Gaussian fluctuations for β ensembles[END_REF]. Later Ryckman ([23] Section 4) used a version of the deformed Verblunsky coefficients in the proof of its joint asymptotic laws.

To be more precise and to explain the interest of our approach, we now describe our model. For n ≥ 1, β > 0 and δ ∈ C such that Re δ > -1/2, we consider a distribution CJ (n) β,δ on the set of probability measures on the unit circle T, supported by n points. This family of distribution generalizes the Circular Jacobi Ensemble (the notation CJ comes from this fact), and it can be defined as follows. If the random measure µ = n j=1 π j δ e iθ j , for θ j ∈ [0, 2π), has the distribution CJ (1e -iθj ) δ (1e iθj ) δ , where c

(n) δ,β > 0 is a normalization constant. • The weights (π 1 , . . . , π n ) follow a Dirichlet law of parameter β ′ on the simplex π 1 + • • • + π n = 1, π j > 0, where β ′ := β/2 (we conserve this notation β ′ in all the sequel of the paper). • The tuples (θ 1 , . . . , θ n ) and (π 1 , . . . , π n ) are independent. Besides, when U is an unitary matrix and e 1 a cyclic vector for U , then the spectral measure µ of the pair (U, e 1 ) is defined as the unique (probability) measure on T such that e 1 , U j e 1 = T z j µ(dz) j ∈ Z .

When δ = 0, it was proved in [START_REF] Killip | Matrix models for circular ensembles[END_REF] that the distribution of the spectral measure of the pair (U, e 1 ) where U is randomly sampled from U(n) according to the Haar measure and e 1 is a fixed vector of C n , for instance (1, 0, . . . , 0), is precisely CJ (n) 2,0 . For δ = 0 and β > 0, these authors found a model of random unitary matrices such that the spectral measure has the distribution CJ (n) β,0 . In [START_REF] Bourgade | Circular Jacobi ensembles and deformed Verblunsky coefficients[END_REF], we gave a model corresponding to the case δ = 0. All these constructions rely on the theory of orthogonal polynomials on the unit circle (OPUC) that we recall now.

From the linearly independent family of monomials {1, z, z 2 , . . . , z n-1 } in L 2 (T, µ), we construct an orthogonal basis Φ 0,n , . . . , Φ n-1,n of monic polynomials by the Gram-Schmidt procedure. The n th degree polynomial obtained this way is

Φ n (z) = Φ n,n (z) = n j=1 (z -e iθj ),
i.e. the characteristic polynomial of U . The Φ k 's (k = 0, . . . , n) obey the Szegö (or Schur) recursion relation:

Φ j+1,n (z) = zΦ j,n (z) -ᾱj Φ * j,n (z) where Φ * j,n (z) = z j Φ j,n (z -1
) . The coefficients α j (j ≥ 0) are called Schur or Verblunsky coefficients and satisfy the condition

α 0 , • • • , α n-2 ∈ D := {z ∈ C : |z| < 1} and α n-1 ∈ T.
There is a bijection between this set of coefficients and the set of spectral probability measures ν (Verblunsky's theorem). We can write the orthogonal polynomials with the help of a new system of functions built from the Verblunsky coefficients. Setting

y k (z) = z - Φ k+1,n (z) Φ k,n (z) = ᾱk Φ * k,n (z) Φ k,n (z)
, (k = 0, . . . , n -1) ,

we have y 0 (z) = ᾱ0 and the following decomposition:

Φ k,n (z) = k-1 j=0 (z -y j (z)) , k = 1, . . . , n .
If γ j := y j (1), we get

Φ k,n (1) = k-1 j=0 (1 -γ j ) , k = 1, . . . n (1.2) and in particular det(I -U ) = Φ n (1) = Φ n,n (1) = n-1 j=0 (1 -γ j ) . (1.3)
Note that the definition implies |γ k | = |α k |, and in particular |γ n-1 | = 1. In the sequel, following [START_REF] Bourgade | Circular Jacobi ensembles and deformed Verblunsky coefficients[END_REF], we refer to the γ j 's as the deformed Verblunsky coefficients.

In [START_REF] Bourgade | Circular Jacobi ensembles and deformed Verblunsky coefficients[END_REF], it is proven that for µ following the distribution CJ

(n)
β,δ , the coefficients (γ j ) 0≤j≤n-1 are independent (note that in general it is not true for the classical Verblunsky coefficients, except if δ = 0) and their distributions are explicitly computable. More precisely, for r > 0 and δ ∈ C such that r + 2Re δ + 1 > 0, let g (δ) r be the density on the unit disc D proportional to

1 -|z| 2 r-1 (1 -z) δ (1 -z) δ
and let λ (δ) be the density on the unit circle U proportional to

(1 -z) δ (1 -z) δ .
Then, for j < n -1, γ j has density g (δ)

β ′ (n-j-1) and γ n-1 has density λ (δ) . Note that the law of γ n-1 does not depend on n and β. The values of the normalization constants can be easily deduced from the computation of integrals which are collected in the appendix of the present paper.

Let us now explain why we choose to focus on the value at 1 of the characteristic polynomial (in addition to its major role in the number theory connections mentioned earlier). Note that in the case δ = 0 (which corresponds to the classical Circular Jacobi Ensemble, and in particular the Circular Unitary Ensemble for β = 2), the law of Φ n (z) does not depend on z ∈ T, since the distribution of the eigenvalues of U is invariant by rotation. On the other hand, for δ = 0, the point 1 plays an important role since it is a singularity of the potential. It is then classical to study the behavior of Φ n (1) in the large n asymptotics (see for example [START_REF] Keating | Random matrix theory and ζ(1/2 + it)[END_REF], [START_REF] Hughes | On the Characteristic Polynomial of a Random Unitary Matrix[END_REF], [START_REF] Killip | Gaussian fluctuations for β ensembles[END_REF], [START_REF] Ryckman | Linear statistics of point processes via orthogonal polynomials[END_REF]). Notice that all these authors consider the case δ = 0.

Our extension to a study of the array {Φ k,n (1), k ≤ n} has its own interest as a study of characteristic polynomials. It comes from the following remark. From a measure µ carried by n points, one can also define a n × n unitary matrix U n , called GGT by Simon [START_REF] Simon | CMV matrices: Five years after[END_REF] section 10, and which is the matrix of the linear application h on L 2 (T, µ) given by h(f )(z) = zf (z), taken in a basis of orthonormal polynomials with respect to µ.

If 1 ≤ k ≤ n, one can denote by G k (U n ) the k × k topleft submatrix of U (which is not unitary in general).
Then it is known (see for example Forrester [START_REF] Forrester | Log-gases and random matrices[END_REF] Prop. 2.8.2, or Simon [START_REF] Simon | CMV matrices: Five years after[END_REF] proof of Prop. 3.1) that one has for all k ∈ {1, 2, . . . , n}:

Φ k,n (1) = det(I k -G k (U n )) (1.4)
For other aspects of this model see [START_REF] Bourgade | Ewens Measures on Compact Groups and Hypergeometric Kernels[END_REF].

Most of the results we will obtain on the process {Φ ⌊nt⌋,n (1), t ∈ [0, 1]} will in fact concern its logarithm. Note that even when Φ k,n (1) = 0, its complex logarithm is not obvious to define rigorously, since its imaginary part is a priori given only modulo 2π. However, there is a natural way to deal with this issue, which is described in the appendix. One can then fully justify the following formula: β,δ . We study asymptotic properties of this determinant as n → ∞ under essentially two regimes:

(1.5) log Φ k,n (1) = k-1 j=0 log(1 -γ j ), when Φ k,n (1)
• First regime: δ is fixed and Re δ > -1/2 (hence, this regime includes the case δ = 0).

• Second regime: δ = β ′ dn with Re d > 0. Some of the results proved here were already announced in [START_REF] Bourgade | The characteristic polynomial on compact groups with Haar measure: some equalities in law[END_REF]. When β = 1, 2, 4, the independence of the random variables Φ k+1,n /Φ k,n k = 0, . . . , n -1 and the identification of their distributions are strongly related to the results of Neretin ([18] Corollary 2.1). In that framework, they can be carried easily on models of matrix balls via his Proposition 2.3. Actually our results may be extended to all models where a remarkable separation of variables occur (see [START_REF] Yu | Matrix analogues of the B-function, and the Plancherel formula for Berezin kernel representations[END_REF]). More precisely the paper is organized as follows.

In Section 2 we study the variables log Φ ⌊nt⌋,n (1) for both regimes and prove that their expectations converge to some explicit deterministic function of t, and then we study the fluctuations of (log Φ ⌊nt⌋,n (1)) as a stochastic process on the space of càdlàg R 2 -valued functions. It appears that in the first regime, one has to distinguish between the case 0 < t < 1 and t = 1; at t = 1 some transition is occuring since the normalization is changed. Indeed, anticipating the notation of next section where we write log Φ ⌊nt⌋,n (1) -E log Φ ⌊nt⌋,n (1) = ξ n (t) + iη n (t) and

ζ n (t) = ξ n (t) η n (t)
, we shall prove that in the first regime, {ζ n (t); t < 1} converges to some explicit Gaussian diffusion {ζ 0 d (t); t < 1} while ζ n (1)/ √ n converges to a Gaussian random variable which is independent of the diffusion {ζ 0 d (t); t < 1}. In the second regime, {ζ n (t); 0 ≤ t ≤ 1} converges to some explicit Gaussian diffusion and there is no normalization to perform.

Section 3 is devoted to the establishing Large Deviation Principle (LDP) for the distributions of the real and imaginary parts of log Φ ⌊nt⌋,n (1) as a two dimensional random vector with values in the Skorokhod space endowed with the weak topology. We focus our study in the first regime on the case δ = 0. Again in the first regime the case t = 1 is playing a special role and is not included. From the contraction principle we are then able to deduce LDP for the marginals at fixed time. Our approach is standard: we first compute the normalized cumulant generating function, compute its limit as well as its dual transform, and end with exponential tightness.

In Section 4 we discuss the connections between the results of Section 3 and existing results on the LDP for the empirical spectral distribution for the circular Jacobi ensemble.

Section 5 gathers in appendix some properties of remarkable functions and densities used in the proofs, in particular the Gamma function Γ and the Digamma function Ψ. For z ∈ C\R -, we take the notation:

ℓ(z) := log Γ(z) ; Ψ(z) = Γ ′ (z) Γ(z) .
The determination of the logarithm in ℓ is chosen in the unique way such that ℓ is holomorphic on C\R -and real on R * + . All along the paper, we use the entropy function J defined by :

(1.6) J (u) =      u log u -u + 1 if u > 0 1 if u = 0 +∞ if u < 0
and its primitive

F (t) = t 0 J (u) du = t 2 2 log t - 3t 2 4 + t, (t ≥ 0) . (1.7)
When the arguments of J or of F are complex, we choose the principal determination of the logarithm.

Convergence and fluctuations

Let, for Re d > 0 and 0 < t ≤ 1,

F d (t) = log(1 + 2Re d) -log(1 + d) -log(1 -t + 2Re d) + log(1 -t + d) (2.1) E d (t) = J (1 + 2Re d) -J (1 + 2Re d -t) -J (1 + d) + J (1 + d -t) .
In this section, we are interested in the process

ζ n (t) = ξ n (t) η n (t)
where we have written log Φ ⌊nt⌋,n (1) -E log Φ ⌊nt⌋,n (1) = ξ n (t) + iη n (t) .

As a consequence of the result just below, E log Φ ⌊nt⌋,n (1) is finite, and then ζ n (t) is well-defined.

2.1. Convergence to a deterministic limit.

Theorem 2.1.

(1) In the first regime, i.e. for fixed δ, and for n going to infinity,

E log Φ n (1) = δ β ′ log n + C + o(1) (2.2)
where C is a constant, and for 0 < t < 1,

E log Φ ⌊nt⌋,n (1) = - δ β ′ log(1 -t) + o(1) . (2.3)
(2) In the second regime, i.e. δ = β ′ dn, and for 0 < t ≤ 1, we have

lim n→∞ E log Φ ⌊nt⌋,n (1) -nE d ⌊nt⌋ n = 1 2 - 1 β F d (t) , (2.4)
uniformly in t.

(3) In the first regime, we have

lim n→∞ 1 log n Cov ζ n (1) = 1 β I 2 , (2.5)
where I 2 denotes the 2 × 2 identity matrix, and for 0 < t < 1,

lim n→∞ Cov ζ n (t) = t 0 Z 0 s ds, (2.6)
where

Z 0 t := 1 β(1 -t) I 2 .
(4) In the second regime, we have for 0 < t ≤ 1:

lim n→∞ Cov ζ n (t) = t 0 Z d s ds (2.7)
where

Z d t = 1 β ′      1 1 -t + 2Re d -Re 1 2(1 -t + d) Im 1 2(1 -t + d) Im 1 2(1 -t + d) Re 1 2(1 -t + d)      . (2.8) (5) In the second regime, sup t∈[0,1] 1 n log Φ ⌊nt⌋,n (1) -E d (t) -→ n→∞ 0 (2.9)
in L 2 , and then in probability.

Proof. Proof of (1) and [START_REF] Borot | Asymptotic expansion of β matrix models in the one-cut regime[END_REF].

Taking expectations in (5.19) and summing up (5.16), we have, for 1 ≤ m ≤ n, (2.10)

E log Φ m,n (1) = n k=n-m+1 Ψ β ′ (k -1) + 1 + δ + δ -Ψ β ′ (k -1) + 1 + δ .
If in (2.10), we keep only the first two terms from the Abel-Plana formula (5.20), we get, for m ≤ n -1, or in the second regime, for m = n large enough,

n n-m Ψ β ′ (s -1) + 1 + δ + δ -Ψ β ′ (s -1) + 1 + δ ds + 1 2 Ψ β ′ (n -1) + 1 + δ + δ - 1 2 Ψ β ′ (n -m -1) + 1 + δ + δ - 1 2 Ψ β ′ (n -1) + 1 + δ + 1 2 Ψ β ′ (n -m -1) + 1 + δ . (2.11)
(Note that in the first regime and for m = n, we would get the terms Ψ -β ′ + 1 + δ Ψ -β ′ + 1 + δ + δ , which are not always well-defined). Integrating Ψ = ℓ ′ , we get the following expression:

L := ℓ(β ′ (n -1) + δ + δ) -ℓ(β ′ (n -m -1) + δ + δ) (2.12) -ℓ(β ′ (n -1) + δ) + ℓ(β ′ (n -m -1) + δ), where ℓ(x) := 1 β ′ ℓ(x + 1) + 1 2 Ψ(x + 1)
. From the Binet formula we have, for x > -1,

ℓ(x) = 1 β ′ x log x + 1 β + 1 2 log x - x -1 β ′ + r 1 (x) (2.13)
where

r 1 (x) = 1 β ′ ∞ 0 f (s)[e -sx -e -s ] ds + 1 2 ∞ 0 e -sx 1 2 -sf (s) ds . (2.14) Now, set I 1 (n, m; δ) = I n -1 + δ + δ β ′ -I n -m -1 + δ + δ β ′ (2.15) -I n -1 + δ β ′ + I n -m -1 + δ β ′ and I 2 (n, m; δ) = 1 β + 1 2 J 2 (n, m; δ) with J 2 (n, m; δ) = log n -1 + δ + δ β ′ -log n -m -1 + δ + δ β ′ (2.16) -log n -1 + δ β ′ + log n -m -1 + δ β ′ .
We have:

L = I 1 (n, m; δ) + I 2 (n, m, δ) + R , where R = r 1 (β ′ (n -1) + δ + δ) -r 1 (β ′ (n -m -1) + δ + δ) -r 1 (β ′ (n -1) + δ) + r 1 (β ′ (n -m -1) + δ).
In the sequel, we use several times the trivial estimates (for c fixed and x tending to infinity)

log(x + c) = log x + o(1) ; I(x + c) = I(x) + c log x + o(1) . (2.17)
Let us set m := ⌊nt⌋, and t n = m/n, and let us suppose that we are in the first regime. If 0 < t < 1, we have 1). In the first regime and for t = 1, we have to estimate

I 1 (n, m, δ) = δ β ′ log n -1 + δ β ′ - δ β ′ log n -nt n -1 + δ β ′ + o(1) = - δ β ′ log(1 -t) + o(1), and J 2 (n, m, δ) = o(1), which implies L = - δ β ′ log(1 -t) + R + o(
E log Φ n (1) = E log Φ n-1,n (1) + Ψ 1 + δ + δ -Ψ 1 + δ .
Since the constant C can be modified, it is equivalent to deal with m = n -1 or with m = n. Taking m = n -1 gives for some constants C 1 and C 2 ,

I 1 (n, n -1, δ) = δ β ′ log n -1 + δ β ′ + I δ + δ β ′ -I δ β ′ + o(1) = δ β ′ log n + C 1 + o(1), J 2 (n, n -1, δ) = log δ β ′ -log δ + δ β ′ + o(1) = C 2 + o(1),
which implies, for some constant C 3 ,

L = δ β ′ log n + R + C 3 + o(1).
Let us now assume that we are in the second regime. For n large enough, we check the following estimates, uniform in t ∈ (0, 1]:

I 1 (n, nt n , nβ ′ d) = nE d (t n ) -J 2 (n, nt n , nβ ′ d) + o(1), and J 2 (n, nt n , nβ ′ d) = F d (t n ) + o(1), which implies L = nE d (t n ) + 1 β - 1 2 F d (t n ) + R + o(1).
Hence, (1) and ( 2) are proven, if we check that in any of the previous situations, E log Φ m,n (1)-L and R tend to a constant when n goes to infinity, that this constant is zero, except perhaps in the first regime for t = 1, and that the convergence is uniform in t in the second regime. The first quantity can be expressed in function of the last integral term of the Abel-Plana formula, and the second one comes from the remaining integral term of the Binet formula. First remaining term: Abel-Plana We have to prove the convergence of:

∞ 0 g(n -m + iy) -g(n + iy) -g(n -m -iy) + g(n -iy) e 2πy -1 dy,
when n goes to infinity, for

g(z) := Ψ(β ′ (z -1) + 1 + δ + δ) -Ψ(β ′ (z -1) + 1 + δ),
the limit being zero, except in the first regime for t = 1. Moreover, in the second regime, we want to check that the convergence is uniform with respect to t, i.e. with respect to m ∈ {1, . . . , n}. Note that in the second regime, the function g depends on n via δ = ndβ ′ . It is then sufficient to check the following result: for any sequence

(k n ) n≥1 of integers such that 1 ≤ k n ≤ n, ∞ 0 g(k n + iy) -g(k n -iy) e 2πy -1 dy
is well-defined for all n ≥ 1, and tends to zero when n goes to infinity, in any case and uniformly with respect to the sequence (k n ) n≥1 in the second regime, and when (k n ) n≥1 tends to infinity in the first regime. Indeed, we get the desired result by taking successively k n = n and k n = nm. Note that in the first regime for t = 1, we need to take k n = 1 (m = n -1), which gives an integral independent of n, possibly different from zero. Now, we have

g(k n ± iy) = Ψ(A ′ ± β ′ iy) -Ψ(A 0 + iB 0 ± β ′ iy), A 0 , A ′ > 0, B 0 ∈ R being given by A 0 + iB 0 = β ′ (k n -1) + 1 + δ and A ′ = β ′ (k n -1) + 1 + δ + δ.
In any case, A 0 and A ′ tend to infinity with n, since in the first regime, k n goes to infinity, and in the second regime, it is the case for the real part of δ. Moreover, in the second regime, A and A ′ are greater than nβ ′ Re d, independently from the sequence (k n ) n≥1 . Hence, it is sufficient to check that

∞ 0 Ψ(A + iB + iC) -Ψ(A + iB -iC) e 2πC/β ′ -1 dC
is well-defined for A > 0 and B ∈ R, and tends to zero, uniformy in B, for A → ∞. From (5.10) we have the following:

-i (Ψ(A + iB + iC) -Ψ(A + iB -iC)) = 2C ∞ k=0 1 (A + k + iB) 2 + C 2 .
To estimate the sum of this series, we first apply the crude estimates |z| ≥ |Im z| and |z| ≥ |Re z| to the denominator:

|(A + k + iB) 2 + C 2 | ≥ 2|B||A + k| |(A + k + iB) 2 + C 2 | ≥ |(A + k) 2 -B 2 + C 2 |.

The first inequality implies, for

A + k ≤ 2|B|, |(A + k + iB) 2 + C 2 | ≥ (A + k) 2 ,
and the second implies, for A + k > 2|B|,

|(A + k + iB) 2 + C 2 | ≥ 3 4 (A + k) 2 + C 2 .
Hence,

∞ k=0 1 (A + k + iB) 2 + C 2 ≤ 4 3 ∞ k=0 1 (A + k) 2 ≤ 4 3 1 A 2 + 1 A .
Therefore,

∞ 0 |Ψ(A + iB + iC) -Ψ(A + iB -iC)| e 2πC/β ′ -1 dC ≤ 4 3 1 A 2 + 1 A ∞ 0 2C e 2πC/β ′ -1 dC,
where the last integral is finite and does not depend on A and B.

Second remaining term: Binet The terms involved in the expression of R are of the form:

R 1 (x, y) = ∞ 0 f (s) e -sx -e -sy ds (2.18) and R 2 (x, y) = ∞ 0 1 2 -sf (s) e -sx -e -sy ds, (2.19)
where x = β ′ (nm -1) + α and y = β ′ (n -1) + α with successively α = δ + δ and α = δ. If t < 1 or if we are in the second regime, the real parts of x and y tend to infinity with n: moreover, the convergence is uniform in t in the second regime. Then, as in [START_REF] Rouault | Asymptotic behavior of random determinants in the Laguerre, Gram and Jacobi ensembles[END_REF], the dominated convergence theorem allows to conclude that R 1 (x, y) and R 2 (x, y) tend to zero, uniformly in t in the second regime. If t = 1 and if we are in the first regime, then y tends to infinity and x = α does not depend on n, which implies that R 1 (x, y) and R 2 (x, y) are still converging when n goes to infinity.

Proof of (3) and ( 4): Computation of covariances. By independence of the variables γ j , we can sum up variances or covariances issued from (5.18). We can then prove the announced result in the same way as (1) and ( 2), using the Abel-Plana summation and the Binet formula again, with Ψ replaced by Ψ ′ : we omit the detail.

Proof of (5): the process (ζ n (t)) 0≤t≤1 is a two-dimensional martingale. Hence, by Doob's inequality:

E sup t∈[0,1] |ζ n (t)| 2 ≤ 4E[|ζ n (1)| 2 ] = 4 Tr(Cov ζ n (1)) -→ n→∞ 1 β ′ 1 0 dt 1 -t + 2Re d = 1 β ′ log 1 + 2Re d 2Re d < ∞.
Moreover, by the uniform convergence (2), sup

t∈[0,1] E log Φ ⌊nt⌋,n (1) -nE d ⌊nt⌋ n -→ n→∞ 1 2 - 1 β sup t∈[0,1] |F d (t)| < ∞,
and one also has

sup t∈[0,1] n E d ⌊nt⌋ n -E d (t) ≤ sup t∈[0,1] |E ′ d (t)| < ∞.
Combining these estimates gives the following L 2 bound:

sup n≥1 E   sup t∈[0,1] log Φ ⌊nt⌋,n (1) -nE d (t) 2   < ∞,
which gives (4) after dividing by n 2 .

2.2. Fluctuations. We now state a theorem about some limiting distributions related to the process {Φ ⌊nt⌋,n (1), t ∈ [0, 1]}. Since it can be shown by adapting the arguments of Killip and Stoiciu [START_REF] Killip | Eigenvalue statistics for CMV matrices: from Poisson to clock via random matrix ensembles[END_REF] for similar results, we omit the proof of our theorem, except for the last part. Let D T and D be the space of càdlàg R 2 -valued functions on [0, T ] and [0, 1) respectively, starting from zero. The next theorem is about some limiting distributions related to the process {ζ n (t), t ∈ [0, 1]} or {ζ n (t), t < 1}. More precisely let (Z d t ) 1/2 denote the positive symmetric square root of Z d t defined in (2.8) and let B t be a standard two dimensional Brownian motion.

Theorem 2.2.

(1) If δ = β ′ dn with Re d > 0 (second regime), then as n → ∞ the process {ζ n (t) ; t ∈ [0, 1]} n converges in distribution in the Skorokhod space D 1 to the Gaussian diffusion {ζ d t ; t ∈ [0, 1]}, solution of the stochastic differential equation:

dζ d t = (Z d t ) 1/2 dB t . (2.20)
(2) If δ is fixed with Re δ > -1/2 (first regime), then the joint law of the process {ζ n (t) ; t < 1} (with trajectories in the Skorokhod space D) and the variable

Θ := log Φ n (1) -δ β ′ log n √ log n
converges, when n goes to infinity, to the joint distribution of {ζ 0 t ; t < 1} and N (0; β -1 ) + i N (0; β -1 ), the process and the two gaussian variables being independent.

Notice that the convergence in law of Θ is an extension of the celebrated Keating and Snaith result [START_REF] Keating | Random matrix theory and ζ(1/2 + it)[END_REF].

Proof. In order to prove (1), and in (2), the convergence of {ζ n (t) ; t < 1} and Θ, taken separately, we apply a version of the Lindeberg-Lévy-Lyapunov criterion, available for convergence of processes ([13] Chap. 3c). For t < 1, or in the second regime, it is enough to prove that

⌊nt⌋-1 k=0 E|A k | 4 -→ n→∞ 0, (2.

21)

where

A k = log(1 -γ k ) -E[log(1 -γ k )].
For t = 1 in the first regime, it is sufficient to check:

1 log 2 n n-1 k=0 E|A k | 4 -→ n→∞ 0. (2.22) Now, E|A k | 4 ≤ 8E(Re A k ) 4 + 8E(Im A k ) 4 ≤ 24 Var 2 (Re log(1 -γ k )) + 24 Var 2 (Re log(1 -γ k )) +8κ 4 (Re log(1 -γ k )) + 8κ 4 (Im log(1 -γ k )),
where κ 4 denotes the fourth cumulant. Using the second and fourth order derivatives of the function Λ introduced in the appendix, we see that E|A k | 4 is a linear combination of terms of the form Ψ ′′′ (r + 1 + α) and Ψ ′ (r + 1 + α)Ψ ′ (r + 1 + α ′ ), for r = β ′ (nk -1) and α, α ′ ∈ {δ, δ, δ + δ}. Now, by (5.9), for Re x > 0, 4 . Hence, all the terms involved in the expression of E|A k | 4 are dominated by (n-k) -2 , and they are dominated by n -2 in the second regime or for t < 1. Hence,

Ψ ′ (x) = 1 x + O 1 (Re x) 2 and Ψ ′′′ (x) = 2 x 3 + O 1 (Re x)
⌊nt⌋-1 k=0 E|A k | 4
is dominated by 1/n, except in the first regime for t = 1, in which case it is bounded. This shows the desired results (2.21) and (2.22).

In order to prove the convergence of the joint distribution in (2), we can follow the scheme of [START_REF] Rouault | Asymptotic behavior of random determinants in the Laguerre, Gram and Jacobi ensembles[END_REF] p.209. We take 0 < t 0 < t 1 < 1. From the above results, log Φ ⌊nt1⌋,n (1)

√ log n → 0 (2.23) in probability, so that log Φ n (1) -log Φ ⌊nt1⌋,n (1) -δ β ′ log n √ log n → N (0; β -1 ) + iN (0; β -1 ),
the two gaussian variables being independent. Now, for n large enough,

log Φ n (1) -log Φ ⌊nt1⌋,n (1) = n-1 k=⌊nt1⌋ (1 -γ k ),
is independent of {ζ n (t) ; t ≤ t 0 }, which is function of (γ k ) k<nt0 (recall that the variables γ k are independent). Since, {ζ n (t) ; t ≤ t 0 } tends in law to {ζ 0 t ; t ≤ t 0 } (as a process with trajectories in D t0 ), we deduce that

{ζ n (t) ; t ≤ t 0 }; log Φ n (1) -log Φ ⌊nt1⌋,n (1) -δ β ′ log n √ log n ,
tends in law to {ζ 0 t ; t ≤ t 0 }; N (0; β -1 ) + iN (0; β -1 ) , {ζ 0 t ; t ≤ t 0 } being independent of N (0; β -1 ) + iN (0; β -1 ). Using again (2.23), we deduce that

{ζ n (t) ; t ≤ t 0 }; log Φ n (1) -δ β ′ log n √ log n
has the same limiting distribution. Taking t 0 → 1 gives part (2) of the theorem.

Large deviations

3.1. Notation and main statements. Throughout this section, we use the standard notation of [START_REF] Dembo | Large Deviations Techniques and Applications[END_REF]. In particular we write LDP for Large Deviation Principle. We say that a sequence (P n ) n∈N of probability measures on a Polish space X satisfies a LDP with speed a n (going to infinity with n) and rate function

I iff I : X → [0, ∞] is lower semicontinuous and if For every open set O ⊂ X , lim inf n 1 a n log P n (O) ≥ -inf O I , For every closed set F ⊂ X , lim inf n 1 a n log P n (F ) ≥ -inf F I .
The rate function is good if its level sets are compact. Moreover, if X n are X random variables distributed according to P n , we say that the sequence (X n ) satisfies the LDP if the sequence (P n ) satisfies the LDP. The reader may have some interest in consulting [START_REF] Dette | Asymptotic properties of the algebraic moment range process[END_REF] and mainly [START_REF] Rouault | Asymptotic behavior of random determinants in the Laguerre, Gram and Jacobi ensembles[END_REF] where a similar method is used for a different model.

For T ≤ 1, let M T stand for the space of signed measures on [0, T ] and let M < be the subspace of M 1 consisting of measures whose support is a compact subset of [0, 1). We endow D × D with the weak topology σ(D × D, M < × M < ). So, D × D is the projective limit of the family, indexed by T < 1, of the topological spaces

(D T × D T , σ(D T × D T , M T × M T )).
Let V ℓ (resp. V r ) be the space of left (resp. right) continuous R-valued functions with bounded variations. We put a superscript T to specify the functions on [0, T ]. There is a bijective correspondence between V T r and M T : -for any v ∈ V T r , there exists a unique µ ∈ M T such that v = µ([0, •]); we denote it by v , -for any

µ ∈ M T , v = µ([0, •]) is in V T r . For v ∈ D, let v = va +
vs be the Lebesgue decomposition of the measure v in absolutely continuous and singular parts with respect to the (vectorial) Lebesgue measure. The measure va can then be identified with its density. Now, for ξ, η ∈ R, let us define:

H a (ξ, η) := -ξ -log(2 cos η -e ξ ) , (3.1)
if |η| < π/2 and 2 cos ηe ξ > 0, and H a (ξ, η) = ∞ otherwise. For ϕ, ψ ∈ V r and T ≤ 1, let us denote:

(3.2) H 0 (T, ϕ, ψ) :=      T 0 (1 -τ )H a ( φa (τ ), ψa (τ )) dτ + T 0 (1 -τ )d(-φs )(τ )
if ψs = 0 andφs is a positive measure , +∞ otherwise. Theorem 3.1.

(1) For every T < 1, the sequence

{n -1 Re log Φ ⌊nt⌋,n (1)
H d (T, ϕ, ψ) = H 0 (T, ϕ, ψ) -2(Re d)ϕ(T ) -2(Im d)ψ(T ) -C d (T ) (3.3) with C d (T ) = F (1 + d) -F (1 -T + d) + F (1 + d) -F (1 -T + d) -F (1 + 2Re d) + F (1 -T + 2Re d) -F (1) + F (1 -T ) .
We give now a result on the marginal at time T fixed. It is obtained by applying the contraction principle to the mapping (ϕ, ψ) → (ϕ(T ), ψ(T )) .

In all the sequel of the paper, we consider either the CβE (n) ensemble (first regime for δ = 0), or the CJ

h d (T, ξ, η) = h 0 (T, ξ, η) -2(Re d)ξ -2(Im d)η -C d (T ), (3.5)
which allows to compute easily h d when h 0 is known.

For the two coordinates, separately, we have the following known result, which comes from formula (C.5) in [START_REF] Hughes | On the Characteristic Polynomial of a Random Unitary Matrix[END_REF]. (1) The sequence {n -1 Re log Φ n (1)} n satisfies the LDP in R with speed n 2 and rate function given by the dual (Legendre) of the function: It is finite on (-π/2, π/2), and infinite otherwise.

s → (1 + s) 2 2 log(1 + s) -1 + s 2 2 log 1 + s 2 - s 2 4 log (
We now give the precise behavior of the first coordinate, in the particular case where d is real. Theorem 3.4.

(1) For (d = 0, T < 1), or (d > 0, T ≤ 1), the sequence {n -1 Re log Φ n (T )} n satisfies the LDP in R with speed β ′ n 2 with good rate function

h d (T, •, 0). (2) Let ξ T := J (T ) -1 -J 1+T 2 + J 1-T 2 ≤ 0. (a) If ξ ∈ [ξ T , T log 2) the equation J (1 + γ) -J (1 -T + γ) -J 1 + γ 2 + J 1 -T + γ 2 = ξ (3.6)
has a unique solution γ and we have

h 0 (T, ξ, 0) = γξ -L 0 (T, γ, 0) (3.7)
where

L 0 (T, γ, 0) := F (1 + γ) -F (1 -T + γ) -F (1 -T ) + F (1) -2F 1 + γ 2 + 2F 1 -T + γ 2 (3.8) (b) If ξ < ξ T , then h 0 (T, ξ, 0) = h 0 (T, ξ T , 0) + (1 -T )(ξ T -ξ) . (3.9) (c) If ξ ≥ T log 2, then h 0 (T, ξ, 0) = ∞.
(3) For (d > 0, T ≤ 1) or (d = 0, T < 1), the function ξ → h d (T, ξ, 0) is the dual function of γ → L d (T, γ, 0) := L 0 (T, γ + 2d, 0) -L 0 (T, 2d, 0) .

3.2.

Proof of Theorem 3.1. We compute the normalized cumulant generating function, find its limit, perform the dual transform, study the exponential tightness and eventually prove that the LDP is satisfied. From (5.12) and (5.13) we see that

CJ β,δ [(1 -γj ) z (1 -γ j ) z ] CJ β,0 (1 -γj ) (z+δ) (1 -γ j ) (z+ δ) = c r,δ c r,0 (3.10)
for every j < n -1 and z such that 2Re (δ + z)) > -1, where r = β ′ (nj -1). The RHS of (3.10) does not depend on z. It should then be clear that we can reduce the case Re d > 0 to the case d = 0. The above shift in the argument of the generating function provides the linear term -2(Re d)ϕ(T ) -2(Im d)ψ(T ) in the rate function, and the RHS of (3.10) gives the constant -C d (T ).

• First step: The normalized cumulant generating function. Lemma 3.5. Let T < 1.

(1) Let us assume d = 0. (a) For every path (x(τ ), y(τ

)) τ ∈[0,T ] ∈ V T ℓ ×V T ℓ such that x(τ )+1-τ > 0 on [0, T ], set 2z(τ ) := x(τ ) + iy(τ ) and Λ 0 (T, x, y) := (3.11) T 0 (J (1 -τ + x(τ )) -J (1 -τ + z(τ )) -J (1 -τ + z(τ )) + J (1 -τ )) dτ .
Then we have:

(3.12) lim n→∞ 1 β ′ n 2 log E exp nβ ′ Re T 0 (x(τ ) -iy(τ )) d log Φ ⌊nτ ⌋ (1)
= Λ 0 (T, x, y) (b) In particular, when x(.) ≡ s and y(.) ≡ t, if we set L 0 (T, s, t) := Λ 0 (T, x, y) (which generalizes the notation L 0 (T, γ, 0) introduced in Theorem 3.4), we have for every s > -(1 -T ) Proof.

L 0 (T, s, t) = F (1 + s) -F (1 -T + s) + F (1) -F (1 -T ) (3.13) -F (1 + z) + F (1 -T + z) -F (1 + z) + F (1 -T + z) where 2z = s + it. ( 2
(1)(a) Let us set x(τ ) = y(τ ) = z(τ ) := 0 for τ > T , and τ j := (j + 1)/n for j = 0, . . . , n -1. One has:

E exp nβ ′ Re T 0 (x(τ ) -iy(τ )) d log Φ ⌊nτ ⌋ (1) = E exp   nβ ′ Re   n-1 j=0 (x (τ j ) -iy(τ j )) log(1 -γ j )     = n-1 j=0 E exp (nβ ′ Re [(x(τ j ) -iy(τ j )) log(1 -γ j )])
and using (5.14), we get:

log E exp nβ ′ Re T 0 (x(τ ) -iy(τ )) d log Φ ⌊nτ ⌋ (1) = n-1 j=0 [ℓ(nβ ′ (1 -τ j + x(τ j )) + 1) + ℓ(nβ ′ (1 -τ j ) + 1) -ℓ(nβ ′ (1 -τ j + z(τ j )) + 1) -ℓ(nβ ′ (1 -τ j + z(τ j )) + 1)] . (3.16)
Now, the Binet formula (5.1) yields:

ℓ(u + 1) = log(u) + log Γ(u) = (u + 1/2) log u -u + 1 + ∞ 0 f (s)[e -su -e -s ]ds .
If we apply four times this formula in order to estimate the term indexed by j < n in (3.16), the contribution of the term u is 0, the contribution of the term log u is log(1τ j ) + log(1τ j + x(τ j ))log(1τ j + z(τ j ))log(1τ j + z(τ j )) , the contribution of the term u log u is proportional to β ′ n with coefficient

J (1 -τ j ) -J (1 -τ j + z(τ j )) -J (1 -τ j + z(τ j )) + J (1 -τ j + x(τ j )) ;
dividing by β ′ n 2 and performing Riemann sums gives the integral in (3.11). The remaining part is a sum of bounded terms which is negligible with respect to n 2 .

(1)(b) The equality (3.13) is obvious by integration.

(2) To get the expression corresponding to Re d > 0 we just use (3.10).

• Second step ; the Legendre duality.

It will be convenient to perform a time-change in (3.11), setting

x(τ ) = (1 -τ )X(τ ) ; y(τ ) = (1 -τ )Y (τ ) ; z(τ ) = (1 -τ )Z(τ ) . (3.17)
The above expression (3.11)

) of Λ 0 becomes Λ 0 (T, x, y) = T 0 (1 -τ )L(X(τ ), Y (τ ))dτ (3.18) where L(X, Y ) = J (1 + X) -J (1 + Z) -J (1 + Z) (3.19) i.e. L(X, Y ) = (1 + X) log(1 + X) + Y arctan Y 2 + X -1 + X 2 log 1 + X 2 2 + Y 2 4 . (3.20)
Looking for the Legendre dual, we see that for |η| < π/2 and e ξ < 2 cos η, the supremum

L ⋆ (ξ, η) = sup X,Y Xξ + Y η -L(X, Y ) is achieved in (X, Y ) satisfying 1 + X 1 + X 2 2 + Y 2 4 = e ξ , Y 2 + X = tan η (3.21) i.e. X = e ξ -cos η cos η -1 2 e ξ , Y = sin η cos η -1
2 e ξ . Note that under the assumption above, X is admissible, i.e. X > -1. One deduces that (cf. (3.1))

L ⋆ (ξ, η) = -ξ -log(2 cos η -e ξ ) = H a (ξ, η). (3.22)
On the other hand, one can check that L ⋆ (ξ, η) is infinite if |η| ≥ π/2 or e ξ ≥ 2 cos η.

One deduces that there exists a recession function:

(ξ, η) → lim κ→+∞ κ -1 L ⋆ (κξ, κη) = -ξ if ξ < 0 and η = 0 , ∞ otherwise .
This function can be used to obtain the rate function H 0 (T, ϕ, ψ) when the measures φs and ψs are not zero. By using the same methods as in [START_REF] Rouault | Asymptotic behavior of random determinants in the Laguerre, Gram and Jacobi ensembles[END_REF], one deduces

H 0 (T, ϕ, ψ) = sup x(.),y(.) T 0 x(τ )d φ(τ ) + y(τ )d ψ(τ ) -Λ 0 (T, x(.), y(.)) = sup X(.),Y (.) T 0 (1 -τ ) X(τ )d φ(τ ) + Y (τ )d ψ(τ ) -L(X(τ ), Y (τ )) dτ = T 0 (1 -τ )H a ( φa (τ ), ψa (τ )) dτ + T 0 (1 -τ )d(-φs )(τ ) ,
where the second equality comes from (3.17 • Third step : exponential tightness. Exponential tightness is not needed for the second argument since it lives in [-π/2, π/2]. For the first, we have | log x| ≤log x + 2 log 2 for x ≤ 2, hence:

P d ( j≤nT -1 | log |1 -y j || ≥ na) ≤ P d ( j≤nT -1 -log(1 -y j ) ≥ n(a -2T log 2)).
Now, for θ < 0 (by Chernov inequality),

P d ( j≤nT -1 -log(1 -y j ) ≥ n(a -2T log 2)) ≤ e n 2 β ′ θ(a-2T log 2) E d |Φ ⌊nT ⌋ (1)| n 2 β ′ θ
so that, taking logarithm and applying (3.13) we get, for θ ∈ (-

(1 -T ) -2Re d, 0) lim sup n→∞ (β ′ n 2 ) -1 log P d ( j≤nT | log |1 -y j | ≥ na) ≤ θ(a -2T log 2) + L d (T, θ, 0).
It remains to let a → ∞ to get the exponential tightness. We remark that when d = 0, the exponential tightness holds only for T < 1.

d dτ (1 -τ ) ∂L ⋆ ∂ φ = 0 d dτ (1 -τ ) ∂L ⋆ ∂ ψ = 0 , (3.23)
and the optimal path is then given by

φ(τ ) = ∂L ∂X γ 1 -τ , ρ 1 -τ , ψ(τ ) = ∂L ∂Y γ 1 -τ , ρ 1 -τ i.e. φ(τ ) = log(1 -τ + γ) - 1 2 log 1 -τ + γ 2 2 + ρ 2 4 , (3.24) ψ(τ ) = arctan ρ 2(1 -τ ) + γ . (3.25)
This path will be admissible if there exist γ and ρ such that

T 0 φ(τ )dτ = ξ , T 0 ψ(τ )dτ = η . (3.26)
When the path is admissible, we have

H a ( φ(τ ), ψ(τ )) = L( φ(τ ), ψ(τ )) = γ 1 -τ φ(τ ) + ρ 1 -τ ψ(τ ) -L γ 1 -τ , ρ 1 -τ and h 0 (T, ξ, η) = γξ + ρη - T 0 (1 -τ )L γ 1 -τ , ρ 1 -τ dτ.
3.4. Comment on Theorem 3.4. In the case d = 0, T = 1, [START_REF] Hughes | On the Characteristic Polynomial of a Random Unitary Matrix[END_REF] proved the LDP by tackling directly the normalized cumulant generating function. This gives an incomplete LDP since there is no steepness in 0. They use a Fourier inversion to take into account the negative side. We see that the function L 0 (1, s, 0) is the limiting n.c.g.f. of log |Φ n (1)|, computed in [START_REF] Hughes | On the Characteristic Polynomial of a Random Unitary Matrix[END_REF] Theorem 3.3. Besides, L 0 (1, 0, t) is the limiting n.c.g.f. of the argument of Φ n (1), computed in [START_REF] Hughes | On the Characteristic Polynomial of a Random Unitary Matrix[END_REF], Theorem 3.4, and

lim t→±∞ L 0 (1, 0, t) t = ± π 2 .
3.5. Proof of Theorem 3.4. In the Im d = 0 case, or the case where d = 0 and T < 1, we could also use the scheme described in Section 3.4. We prefer illustrate the method of contraction, where we will see that the counterpart of the singular contribution is an affine part. To simplify the exposition, we assume the problem one-dimensional (consider only the first component) and

d > 0. Since ∂L d ∂s (T, s, 0) = ∂L 0 ∂s (T, s + 2d, 0) = J (1 + s + 2d) -J (1 -T + s + 2d) -J (1 + s 2 + d) + J (1 -T + s 2 + d) and since J (a + s) -J (b + s) = (a -b) log s + o(1) as s → ∞, we have lim s→∞ ∂L d ∂s (s, 0) = T log 2
which corresponds to the endpoint of the interval allowed for Re log Φ n (T ). On the other side, if d > 0,

lim s↓-(1-T )-2d ∂L d ∂s (T, s, 0) = J (T ) -1 -J 1 + T 2 + J 1 -T 2 =: ξ T .
There is a problem of non-steepness since it is not -∞.

The first coordinate satisfies the LDP with good rate function ξ → inf{h 0 (T, ξ, η)|η} and from (3.4)

inf{h 0 (T, ξ, η)|η} = inf{H 0 (T, ϕ, ψ)|ϕ(T ) = ξ}
From the structure of H 0 and H a , it is clear that this infimum is achieved for ψ(.) = 0 i.e. ρ = 0.

In the case of admissible ϕ, it remains to compute γ. Let us study the mapping γ → ϕ(T, γ) where ϕ(•, γ) is given by (3.24) with ρ = 0. We follow the lines of argument of [START_REF] Rouault | Asymptotic behavior of random determinants in the Laguerre, Gram and Jacobi ensembles[END_REF] pp. 3216-3218. We have

∂ϕ(T, γ) ∂γ = log 1 + γ 1 -T + γ - 1 2 log 1 + γ 2 1 -T + γ 2 > 0 . (3.27) The mapping γ → ϕ(T, γ) is then bijective from [-(1 -T ), ∞) to [ξ T , T log 2).
Fixing ξ ∈ (-∞, T log 2], let us look for optimal ϕ. Let γ > -(1 -T ) (playing the role of a Lagrange multiplier). By the duality property, we have the inequality

(1 -τ )L ⋆ ( φa (τ )) ≥ γ φa (τ ) -(1 -τ )L γ 1 -τ , 0 .
Using (3.2) and (3.22) we get, by integration of the above inequality:

H 0 (T, ϕ, 0) = T 0 (1 -τ )L ⋆ ( φa (τ ))dτ + T 0 (1 -τ )d (-φs ) (τ ) ≥ γϕ a (T ) - T 0 (1 -τ )L γ 1 -τ , 0 dτ + T 0 (1 -τ )d (-φs ) (τ ).
For every ϕ such that ϕ(T ) = ξ, we then have

H 0 (T, ϕ, 0) ≥ γξ - T 0 (1 -τ )L γ 1 -τ , 0 dτ - T 0 (1 -τ + γ)d φs (τ ) ≥ γξ - T 0 (1 -τ )L γ 1 -τ , 0 dτ. (3.28)
We can now distinguish three cases:

• If ξ ∈ [ξ T , T log
2), we choose the path v ξ absolutely continuous and such that

vξ (τ ) = log(1 -τ + γ ξ ) -log 1 -τ + γ ξ 2 ,
where γ ξ is uniquely determined by the condition v ξ (T ) = ξ. This path saturates the infimum and the expression of the action integral is clear.

• If ξ < ξ T , set ε = ξ T -ξ. Plugging γ = -(1 -T ) in (3.28) yields for v such that v(T ) = ξ: H 0 (T, v, 0) ≥ -(1 -T )ξ - T 0 (1 -τ )L -(1 -T ) 1 -τ , 0 = (1 -T )ε + h 0 (T, ξ T , 0)
and this lower bound is achieved for the measure v = v ξT (τ )dτεδ T , since

H a (v ξT ) = h 0 (T, ξ T , 0) , T 0 (1 -τ )ε dδ T (τ ) = (1 -T )ε .
• If ξ = T log 2, make ξ = T log 2 in (3.28). We get, for all γ > -(1 -T ),

h 0 (T, T log 2, 0) ≥ γT log 2 - T 0 (1 -τ )L γ 1 -τ , 0 dτ. (3.29) When γ → ∞, the integral is F (1 + γ) -F (1 -T + γ) + F (1) -F (1 -T ) -2F (1 + γ/2) + 2F (1 -T + γ/2
) which tends to -∞ as -T log γ, so that finally, the RHS (3.29) tends to ∞ and we conclude h 0 (T, T log 2, 0) = ∞.

Connection with the spectral method

It can be interesting to connect the results of the previous section to the results obtained by looking directly at the empirical spectral distribution of the ensembles which are considered. This point of view is also discussed in [START_REF] Hughes | On the Characteristic Polynomial of a Random Unitary Matrix[END_REF].

The LDP for the empirical spectral distribution of the unitary ensemble is given in [START_REF] Hiai | A large deviation theorem for the empirical eigenvalue distribution of random unitary matrices[END_REF]. The rate function is the Voiculescu's logarithmic entropy:

I(µ) = -Σ T (µ) := - T×T log |z -z ′ | dµ(z)dµ(z ′ ). (4.1)
The circular Jacobi unitary ensemble yields also a LDP given in [START_REF] Bourgade | Circular Jacobi ensembles and deformed Verblunsky coefficients[END_REF]. The rate function is x log dx . (4.4) (Notice that there was a mistake in [START_REF] Bourgade | Circular Jacobi ensembles and deformed Verblunsky coefficients[END_REF], fixed in the arXiv version). With our notation, it yields:

I d (µ) = -Σ T (µ) + T Q d (z)dµ(z) + B(d) , (4.2) where
B(d) = F (1 + 2Re d) -F (2Re d) -2Re F (1 + d) + 2Re F (d) + F (1) .
If the mapping µ ∈ M 1 (T) → log(1z) dµ(z) were continuous, we would have by contraction:

h d (1, ξ, η) = inf{I d (µ) | µ ∈ M 1 (T) : T log(1 -z) dµ(z) = ξ + iη} . (4.5)
We conjecture that this formula holds nevertheless, and we prove it in the onedimensional case.

h d (1, ξ, 0) = inf{I d (µ) | T log |1 -z| dµ(z) = ξ} . (4.6)
We use the following interesting result. Proposition 4.2. We have

inf{-Σ T (µ) | µ ∈ M 1 (T) : T log |1 -z| dµ(z) = ξ} = -Σ T (µ a ) (4.7)
where

• the measure µ a ∈ M 1 (T) is defined by

(4.8) dµ a (z) = (1 + a) sin 2 θ/2 -sin 2 (θ a /2) 2π sin(θ/2) 1 (θa,2π-θa) (θ) dθ
where z = e iθ , θ ∈ [0, 2π] and θ a ∈ (0, π) is such that sin θ a /2 = a 1+a . • a is the unique solution of

T log |1 -z| dµ a (z) = ξ. Moreover, T arg(1 -z) dµ a (z) = 0 .
The two propositions mean that to get the non standard mean value ξ for the logarithm of the determinant, the most probable way is to force the random operator to get an empirical spectral distribution close to µ a . The RHS of (4.6) is -Σ T (µ a ) and it remains to prove that it fits with the value

h 0 (1, ξ, 0) = γξ -F (1 + γ) + F (γ) -F (1) + 2F 1 + γ 2 -2F γ 2
where in view of Theorem 3.4 (2), γ is the solution of (3.6) i.e.

J (1 + γ) -J (γ) -J 1 + γ 2 + J γ 2 = ξ . (4.9)
Applying [START_REF] Bourgade | Circular Jacobi ensembles and deformed Verblunsky coefficients[END_REF] formula (5.25)

dθ = 2du cos(θ/2) = 2du √ 1 -u 2 ,
and then

I = 4 1 a/(1+a) log(2u) u 2 -[a/(1 + a)] 2 1 -u 2 du u .
The second change of variable

v = u 2 -[a/(1 + a)] 2 1 -u 2 ⇐⇒ u = v 2 + [a/(1 + a)] 2 1 + v 2 , gives log(2u) = log 2 + 1 2 log v 2 + [a/(1 + a)] 2 - 1 2 log(v 2 + 1), du u = v v 2 + [a/(1 + a)] 2 - v v 2 + 1 dv.
Hence,

I = 4 ∞ 0 log 2 + 1 2 log v 2 + [a/(1 + a)] 2 - 1 2 log(v 2 + 1) • v 2 v 2 + [a/(1 + a)] 2 - v 2 v 2 + 1 dv (4.13) = 2 ∞ -∞ log 2 + 1 2 log v 2 + [a/(1 + a)] 2 - 1 2 log(v 2 + 1) • 1 v 2 + 1 - [a/(1 + a)] 2 v 2 + [a/(1 + a)] 2 dv. Now, for α, β > 0, (4.14) ∞ -∞ α 2 v 2 + α 2 dv = απ, (4.15) ∞ -∞ α 2 log(v 2 + β 2 ) v 2 + α 2 dv = 2απ log(α + β).
The first equality (4.14) is elementary. To get (4.15), one can differentiate the LHS with respect to β, and one obtains, for α = β,

∞ -∞ 2α 2 β (v 2 + α 2 )(v 2 + β 2 ) dv = 2α 2 β β 2 -α 2 ∞ -∞ dv v 2 + α 2 - ∞ -∞ dv v 2 + β 2 = 2πα α + β .
By continuity, the last equality remains true for α = β, and reversing the differentiation gives

∞ -∞ α 2 log(v 2 + β 2 ) v 2 + α 2 dv = 2απ log(α + β) + C(α). Now, to determinate C(α) let us write ∞ -∞ α 2 log(v 2 + β 2 ) v 2 + α 2 dv = α log(β 2 ) ∞ -∞ α v 2 + α 2 dv + α ∞ -∞ α log(1 + v 2 /β 2 ) v 2 + α 2 dv .
The first integral is equal to 2απ log(β) and the second tends to zero for α fixed and β → ∞ by dominated convergence. This gives C(α) = 0 and then (4.15). Expanding (4.13), then using (4.14) and (4.15) gives:

I = 2π 1 + a [(1 + 2a) log(1 + 2a) -(1 + a) log(1 + a) -2a log(2a) + a log(a)] .
Coming back to the definition of I, we obtain:

T log |1 -ζ| dµ a (ζ) = I(1 + 2a) -I(1 + a) -I(2a) + I(a) = E a (1) .
In particular,

T log |1 -ζ| dµ γ/2 (ζ) = ξ,
and the value of a involved in Proposition 4.2 is equal to γ/2. It remains to check that -Σ T (µ γ/2 ) = h 0 (1, ξ, 0). Indeed,

-Σ T (µ γ/2 ) = -Q γ/2 (ζ)dµ γ/2 (ζ) -B(γ/2) = γξ -B(γ/2) = γξ -F (1 + γ) + F (γ) + 2F (1 + γ/2) -2F (γ/2) -F (1) ,
which proves Proposition 4.1.

Remark 4.3. The determination of I could also be viewed as a consequence of the weak convergence of the empirical spectral measure under CJ naβ ′ . Of course the mapping µ → T log |1 -ζ| dµ(ζ) is not continuous, but since the extremal eigenvalues converge to the extreme points of the support of µ a (see [START_REF] Borot | Asymptotic expansion of β matrix models in the one-cut regime[END_REF]), we have

T log |1 -ζ| dµ a (ζ) = lim n→∞ 1 n log |Φ n (1)| = E a (1)
where the last equality comes from (2.9) in Theorem 2.1.

Proof of Proposition 4.2 A possible way consists in studying the maximization of the functional

Σ T (µ) + r T log |1 -z| dµ(z) ,
where r is some Lagrange multiplier.

It is convenient to push-forward this problem to R via the Cayley transformation

z = e iθ = λ + i λ -i so that, for z, z ′ = 1, log |1 -z| = - 1 2 log(1 + λ 2 ) + log 2 log |z -z ′ | = log |λ -λ ′ | - 1 2 log(1 + λ 2 ) - 1 2 log(1 + λ ′2 ) + log 2 .
Hence, one needs to minimize:

Σ R (ν) + 2 R Q(λ) dν(λ), over ν ∈ M 1 (R), where Σ R (ν) = R×R log 1 |λ -λ ′ | dν(λ)dν(λ ′ ) and 2Q(x) = 1 + r 2 log(1 + x 2 ).
This problem can be connected to the research of the equilibrium measure in the Cauchy ensemble, i.e. for the Coulomb gas with probability distribution given by formula (3.124) in Forrester [START_REF] Forrester | Log-gases and random matrices[END_REF]. According to Saff and Totik [START_REF] Saff | of Grundlehren der Mathematischen Wissenschaften[END_REF], this potential Q is admissible (p.27) as soon as r > 0. In that case the minimizer is unique (Theorem 1.3), its support is compact. In the book, an explicit method is presented for a class of potentials satisfying some conditions, in particular if Q is even, differentiable and such that xQ ′ (x) is positive and increasing in (0, ∞). These properties are satisfied here since:

xQ ′ (x) = 1 + r 2 x 2 1 + x 2 . Then ([24] Corollary 1.12 p. 203) the support is S = [-b, b] where b is solution of 2 π 1 0 btQ ′ (bt) √ 1 -t 2 dt = 1 (4.16) i.e. 2 + r π 1 0 b 2 t 2 (1 + b 2 t 2 ) √ 1 -t 2 dt = 1 or 1 0 dt (1 + b 2 t 2 ) √ 1 -t 2 = πr 2(2 + r) (4.17) which gives (make t = (1 + τ 2 ) -1/2 ) ∞ 0 dτ 1 + b 2 + τ 2 = πr 2(2 + r) i.e. b = 2 √ 1 + r r .
To find the extremal measure, we apply Theorem IV.3.1 in [START_REF] Saff | of Grundlehren der Mathematischen Wissenschaften[END_REF], which contains the following result:

Theorem 4.4 (Lubinsky-Saff). Let f be a differentiable even function on [-1, +1], such that sf ′ (s) is increasing for s ∈ (0, 1) and for some

1 < p < 2, f ′ (s)/ √ 1 -s 2 ∈ L p [-1, 1]. Then the integral equation 1 -1 log 1 |x -t| g(t)dt = -f (x) + C f , x ∈ (-1, 1) ,
(where C f is some constant) has a solution of the form

g(t) = L[f ′ ](t) + B f π √ 1 -t 2 (4.18)
where

L[f ′ ](t) = 2 π 2 1 0 √ 1 -t 2 (sf ′ (s) -tf ′ (t)) √ 1 -s 2 (s 2 -t 2 ) ds, (4.19) B f = 1 - 1 π 1 -1 sf ′ (s) √ 1 -s 2 ds .
Here

(bt) = g(t), B f = 0 and sf ′ (s) -tf ′ (t) s 2 -t 2 = b 2 1 + r 2 1 (1 + b 2 t 2 )(1 + b 2 s 2 )
.

Owing to (4.17), we get

bg b (bt) = 1 + √ 1 + b 2 π √ 1 -t 2 (1 + b 2 t 2 ) I [-1,1] (t)
i.e.

g b (x) = 1 + √ 1 + b 2 bπ √ 1 -x 2 b -2 (1 + x 2 ) I [-b,b] (x).
(x) = (x - 1 2 ) log x -x + 1 + ∞ 0 f (s)[e -sx -e -s ] ds (5.1) = (x - 1 2 ) log x -x + 1 2 log(2π) + ∞ 0 f (s)e -sx ds , (5.2)
where the function f is defined by

f (s) = 1 2 - 1 s + 1 e s -1 1 s = 2 ∞ k=1 1 s 2 + 4π 2 k 2 , (5.3)
and satisfies for every s ≥ 0: x + Ψ(x), we have a variation of (5.5):

0 < f (s) ≤ f (0) = 1/12 , 0 < sf (s) + 1 2 < 1 . (5.4) By differentiation (recall that Ψ is the Digamma function Γ ′ /Γ), log x -Ψ(x) = 1 2x + ∞ 0 sf (
log x -Ψ(x + 1) = - 1 2x + ∞ 0 sf (s)e -sx ds (5.6) = ∞ 0 e -sx sf (s) - 1 2 ds .
As easy consequences, we have, for every x > 0,

0 < x (log x -Ψ(x)) ≤ 1 , (5.7) 0 < log x -Ψ(x) - 1 2x ≤ 1 12x 2 . (5.8)
Differentiating again we see that for q ≥ 1, Re x > 0, Ψ (q) (x) = (-1) q-1 (q -1)!x -q + (-1) q-1 ∞ 0 e -sx s q sf (s) + 1 2 ds and then |Ψ (q) (x) -(-1) q-1 (q -1)!x -q | ≤ (Re x) -q-1 q! . (5.9) Another useful formula is

Ψ(z + 1) = -γ - ∞ k=1 1 k + z - 1 k , (5.10) fir z + 1 / ∈ R -.

The density g

(δ) r and some moments related to it. Recall that for r > 0 and δ such that r + 2Re δ + 1 > 0, the density g (δ) r on the unit disc D is given by

g (δ) r (z) = c r,δ (1 -|z| 2 ) r-1 (1 -z) δ (1 -z) δ
where c r,δ is the normalization constant. The following lemma is the key to compute c r,δ and the moments of g (δ) r .

Lemma 5.1. Let s, t, ℓ be complex numbers such that: Re ℓ, Re(s + ℓ + 1), Re(t + ℓ + 1) and Re(s + t + ℓ + 1) are strictly positive. Then, the following identity holds: A proof of this result is given in [START_REF] Bourgade | Circular Jacobi ensembles and deformed Verblunsky coefficients[END_REF]. A first consequence is that c r,δ = π -1 Γ r + 1 + δ , r + 1 + δ r , r + 1 + δ + δ . (5.12)

D (1 -|z| 2 ) ℓ-1 (1 -z) s
A second consequence is that if γ has the density g as soon as all the real parts of the arguments of the gamma functions are strictly positive.

Let us notice that for r = 0 the RHS of (5.13) is the Mellin-Fourier transform of 1γ when γ ∈ T is distributed according to λ (δ) .

In this paper, we need the following computations, in order to deduce the moments of log(1γ). The quantities involved below are all well-defined as soon as s > s 0 , where s 0 is some strictly negative quantity depending on r and δ, and in particular, for (s, t) in the neighborhood of (0, 0), one can write: where the λ j 's are the roots, counted with multiplicity, of the polynomial z → det(zI k -V ), and where in the right-hand side, one considers the principal branch of the logarithm. This definition is meaningful, since by assumption, 1λ j / ∈ R - for all j ∈ {1, . . . , k}. By the continuity of the set of roots of a polynomial with respect to its coefficients, the set E k is open and the function V → log det(I k -V ) defined just above is continuous on E k . In fact, since E k is connected (this is easily checked by tridiagonalizing the matrices), this is the unique way to define the logarithm of det(I k -V ) as a continuous function of V ∈ E k if we assume that it should take the value zero at V = 0. Now, with the notation of the beginning of the paper, the matrix G k (U n ) is a submatrix of the unitary matrix U n , and all its eigenvalues have modulus bounded by 1. If we assume γ 0 , . . . , γ n-1 = 1 (which holds almost surely under CJ is continuous if we take the principal branch of the logarithm, and since U n depends continuously on (γ 1 , . . . , γ n-1 ) ∈ D n-1 × (U\{1}), as it can be checked in [START_REF] Bourgade | Circular Jacobi ensembles and deformed Verblunsky coefficients[END_REF], the map (γ 1 , . . . , γ n-1 ) → log Φ k,n [START_REF] Abramowitz | Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF] is also continuous. These two maps have the same exponential, and one can check that they are both real if the γ j 's are all real. Hence, they are equal, which fully justifies the equation (5.20)

For a proof see [START_REF] Olver | Asymptotics and special functions[END_REF] p. 290.

  , then:• The joint density h (n) δ,β of the law of (θ 1 , . . . , θ n ), with respect to the Lebesgue measure on [0, 2π) n , is given by(1.1) h (n) δ,β (θ 1 , . . . , θ n ) = c (n) δ,β |∆(e iθ1 , . . . , e iθn )| β n j=1

= 1 ,

 1 which occurs almost surely under CJ (n)

,

  Im log Φ ⌊nt⌋,n (1) ; t ∈ [0, T ]} under the CβE (n) measure (i.e. first regime and δ = 0) satisfies in (D T × D T , σ(D T ×D T , M T ×M T )) the LDP with speed β ′ n 2 with good rate function H 0 (T, ϕ, ψ). (2) For Re d > 0 and T ≤ 1, the sequence {n -1 Re log Φ ⌊nt⌋,n (1), Im log Φ ⌊nt⌋,n (1) t ∈ [0, T ]} under the CJ (n) β,β ′ dn measure (second regime) satisfies in (D T × D T , σ(D T × D T , M T × M T )) the LDP with speed β ′ n 2 with good rate function

4 )

 4 ′ dn ensemble for Re d > 0 (second regime). In the first case, we put d = 0. Theorem 3.2. When (Re d > 0, T ≤ 1) or (d = 0, T < 1), the sequence {n -1 Re log Φ ⌊nT ⌋ (1), Im log Φ ⌊nT ⌋ (1) } n satisfies the LDP in R 2 with speed β ′ n 2 with good rate function h d (T, ξ, η) = inf{H d (T, ϕ, ψ) | ϕ(T ) = ξ, ψ(T ) = η} . (3.In particular (cf. (3.3))

Theorem 3 . 3 .

 33 Assume d = 0 and β = 2.

  ) For Re d > 0, the analogues of (3.11) and (3.13) are (3.14) Λ d (T, x, y) = Λ 0 (T, x + 2Re d, y + 2Im d) -Λ 0 (T, 2Re d, 2Im d) , and (3.15) L d (T, s, t) = L 0 (T, s + 2Re d, t + 2Im d) -L 0 (T, 2Re d, 2Im d) , for s > -(1 -T ) -2Re d.

  ) and(3.18), and where the last equality comes from[START_REF] Rockafellar | Integrals which are convex functionals[END_REF] Theorem 5.The value of the constant C d (T ) is obtained owing to (3.13):C d (T ) = -L 0 (T, 2Re d, 2Im d) .

Remark 3 . 6 .3. 3 .

 363 The mean trajectory is obtained when H a ( φ, ψ) ≡ 0 i.e. cos ψ = cosh φ or φ = ψ = 0 Comment on Theorem 3.2. Let us study the variational problem (3.4) issued from the contraction. Using (3.2) and (3.22), we see that the Euler equation is

(4. 3 ) 2 -B(d) = 1 0[

 321 Q d (e iθ ) := -2(Re d) log 2 sin θ (Im d)(θπ) (θ ∈ (0, 2π)) . and (x + 2Re d) log(x + 2Re d) -2Re [(x + d) log(x + d)]] dx + 1 0

Proposition 4 . 1 .

 41 With the notation of Section 3, we have, for d > 0 and for every ξ ∈ [0, log 2)

  Proof of Proposition 4.1 given Proposition 4.2. Let us first suppose that the Proposition 4.1 is true for d = 0. One deduces, from this assumption and the definition (4.2) of I d : inf{I d (µ) | µ ∈ M 1 (T) : log |1 -z|dµ(z) = ξ} = inf{-Σ T (µ) | µ ∈ M 1 (T) : log |1 -z|dµ(z) = ξ} + B(d) -2dξ = h 0 (1, ξ, 0) + B(d) -2dξ = h d (1, ξ, 0), the last equality coming from (3.5) and the fact that B(d) = -C d (1). Hence, it is sufficient to prove Proposition 4.1 for d = 0.

  in the special case d = a (real and positive), we see that, since the corresponding rate function vanishes for the limiting empirical measure µ a , Σ T (µ a ) = T Q a (ζ) dµ a (ζ) + B(a) , (4.10) where Q a (ζ) = -2a log |1 -ζ|. Let us now compute: (4.11) I := 2π 1 + a T log |1 -ζ| dµ a (ζ) .We have, by definition of µ a in (4.8), (θ/2)) sin 2 (θ/2)sin 2 (θ a /2) sin(θ/2) dθ , (4.12) for θ a ∈ [0, π] and sin(θ a /2) = a/(1 + a). The first change of variables u = sin(θ/2) gives

  , we look for the measure µ with support [-b, b] such that for z ∈ (-b, b), dµ(t) = -Q(z) + C. (4.20) Theorem 4.4 is set for b = 1, so we need a scaling. Equation (4.20) becomes, with dµ(x) = g b (x)dx, bg b (bt)dt = -Q(bz) + C ′ . (4.21) Theorem 4.4 can now be applied by taking f (s) = Q(bs), and one has bg b

5 . 1 .

 51 Now, it is enough to apply Theorem I.3.3 of[START_REF] Saff | of Grundlehren der Mathematischen Wissenschaften[END_REF] as follows. The support of the equilibrium measure is S = [-b, b], the measure µ is supported by S, has a finite logarithmic energy and b -b log1 |z-t| dµ(t) + Q(z) is constant for z ∈ S, so µ is the equilibrium measure. To find µ a , it is enough to carry µ back on the circle by the inverse transformation Some properties of ℓ = log Γ and Ψ = ℓ ′ . From the Binet formula (Abramowitz and Stegun[START_REF] Abramowitz | Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF] or Erdélyi et al.[START_REF] Erdelyi | Higher transcendental functions, volume I. Krieger[END_REF] p.21), we have for Re x > 0 ℓ

  s)e -sx ds =

  (1z) t d 2 z = πΓ ℓ , ℓ + 1 + s + t ℓ + 1 + s , ℓ + 1 + t , (5.11)where for the sake of simplicity we use the polygamma symbolΓ a, b, • • • c, d, • • • := Γ(a)Γ(b) • • • Γ(c)Γ(d) • • • .

  we haveE(1γ) a (1γ) b = Γ r + 1 + δ + δ + a + b , r + 1 + δ , r + 1 + δ r + 1 + δ + δ , r + 1 + δ + a , r + 1 + δ + b(5.13)

ΛEE 4 Ψ 4 Ψ

 44 (s, t) := log E exp (2sRe log(1γ) + 2tIm log(1γ)) = = log E exp (Re (2(sit) log(1γ)) = = log E(1γ) s-it (1γ) s+it = = ℓ r + 1 + δ + δ + 2sℓ r + 1 + δ + δ (5.14) -ℓ r + 1 + δ + sitℓ r + 1 + δ + s + it +ℓ r + 1 + δ + ℓ r + 1 + δ .To compute moments we need differentiation. First we have:∂ ∂s Λ(s, t) = 2Ψ r + 1 + δ + δ + 2s -Ψ (r + 1 + δ + s + it) -Ψ r + 1 + δ + sit (5.15) ∂ ∂t Λ(s, t) = iΨ r + 1 + δ + sit -iΨ r + 1 + δ + s + it .The first moment is then:Re log(1γ) = Ψ(r + 1 + δ + δ)log(1γ) = Ψ(r + 1 + δ + δ) -Ψ(r + 1 + δ) . (5.16)Differentiating again (5.15) we get∂ 2 ∂s 2 Λ(s, t) = 4Ψ ′ r + 1 + δ + δ + 2s -Ψ ′ r + 1 + δ + s + it -Ψ ′ r + 1 + δ + sit ∂ 2 ∂t 2 Λ(s, t) = Ψ ′ r + 1 + δ + sit + Ψ ′ r + 1 + δ + s + it (5.17) ∂ 2 ∂s∂t Λ(s, t) = -iΨ ′ r + 1 + δ + s + it + iΨ ′ r + 1 + δ + sitand the second moments areVar Re log(1γ) = Ψ ′ r + 1 + δ + δ -1 ′ r + 1 + δ -1 ′ r + 1 + δ Var Im log(1γ) = 1 4 Ψ ′ r + 1 + δ + 1 4 Ψ ′ r + 1 + δ (5.18) Cov (Re log(1γ), Im log(1γ)) = 1 4i Ψ ′ r + 1 + δ -1 4i Ψ ′ r + 1 + δ .

5. 3 .

 3 Complex logarithm and characteristic polynomial. Let E k be the set of the complex k × k matrices with no eigenvalue on the interval [1, ∞). For V ∈ E k , let us define log det(I k -V ) := k j=1 log(1λ j ),

1 j=0log( 1 -

 11 ), then by (1.2), Φ k,n (1) = 0, and one easily deduces that G k (U n ) ∈ E k , which allows to define log Φ k,n (1) without ambiguity. Now, the map from D n-1 × (U\{1}) to R, given by (γ 1 , . . . , γ n-1 ) → kγ j )

4 .Theorem 5 . 2 .

 452 Abel-Plana summation formula. Let m < n be integers and let g be a holomorphic function on the strip {t ∈ C, n ≤ Re t ≤ m} (i.e. g is continuous on this strip and holomorphic in its interior). We assume that g(t) = o (exp(2π|Im t|)) as Im t → ±∞, uniformly with respect to Re t ∈ [n, m]. Then, + iy)g(n + iy)g(miy) + g(niy) e 2πy -1 dy .

  The sequence {n -1 Im log Φ n (1)} n satisfies the LDP in R with speed n 2 and good rate function given by the dual (Legendre) of the function given by

								2s)
	for s ≥ 0, and by ∞ for s < 0. It vanishes for negative values of the argument, and it is infinite beyond log 2.
	(2) t →	t 2 8	log 1 +	4 t 2 -	1 2	log 1 +	t 2 4	+ t arctan(t/2) .