Michèle Artigue

Michele Cerulli

C.N T D R Of Genova

CONNECTING THEORETICAL FRAMEWORKS: THE TELMA PERSPECTIVE

In this text, we report on a research project developed within the European research team TELMA (Technology Enhanced Learning in MAthematics) of the Kaleidoscope network of excellence created in 2004. We describe the conceptual and methodological tools we have progressively built for allowing productive research collaboration and overcoming the difficulties resulting from the diversity and heterogeneity of our respective theoretical backgrounds. We also show how these tools have contributed to give us a clearer idea of what is needed in terms of theoretical connection and integration in mathematics education, of what seems accessible today and how.

INTRODUCTION

Research in mathematics education does not obey a unified paradigm. On the contrary, it often appears as a field broken into a multiplicity of local communities that develop more or less independently, generating an overflow of conceptual and methodological tools poorly connected. In spite of the multiplicity of international conferences and groups, in spite of evident common trends, exchanges remain often superficial. Even if anyone understands the necessary sensitivity of the educational domain to social and cultural contexts, this situation conveys the negative image of an immature scientific field and does not encourage at considering the results obtained in it as convincing and valuable. Such a situation appears more and more problematic, increasing the attention paid to issues of comparison and connection between theoretical frames, as illustrated for instance by two recent issues of the Zentralblatt für Didaktik der Mathematik (ZDM 2005 Vol. 37(6), ZDM 2006 Vol. 38(1)), the chapter by Cobb in the second NCTM Handbook of Research on Teaching and Learning Mathematics [START_REF] Cobb | Putting Philosophy to Work: Coping With Multiple Theoretical Perspectives[END_REF] or the existence of a working group especially devoted to these issues at the two last conferences of the European Association for Research in Mathematics Education (Bosch, 2006). Research concerning digital technologies does not escape this rule as evidenced for instance by the meta-study [START_REF] Lagrange | Technology and Mathematics Education : A Multidimensional Study of the Evolution of Research and Innovation[END_REF] but, due to the normal ambition of artefact designers to develop tools not restricted to one particular local community and able to 1-2 Joint Meeting PME 32 -NA XXX -2008 migrate from one educational context to another one, researchers in that area are perhaps more sensitive to the problems raised by the current fragmentation of the field.

Within the European research team TELMA, we faced the difficulties generated by this situation when exploring possibilities for collaboration between the six different teams involved. In this paper, we report on the TELMA enterprise which began four years ago and led us to develop specific tools for overcoming these difficulties. We first briefly present the TELMA structure then focus on the conceptual and methodological tools that we have developed. After describing these, we try to show how these tools have contributed to give us a clearer idea of what is needed in terms of theoretical connection and integration in mathematics education, of what seems accessible today and how.

TELMA: AIMS, CHARACTERISTICS AND FIRST STEPS

TELMA (Technology Enhanced Learning in Mathematics) is a sub-structure of the Kaleidoscope European Network of Excellence. It includes six European teams from four different countries (England, France, Greece and Italy), and its main aims is to promote networking and integration among such teams for favouring the development of collaborative research and development projects on the teaching and learning of mathematics with digital technologies. The TELMA teams have a long experience in that area but they live in different educational contexts, the digital technologies they have developed are diverse, ranging from half baked microworlds to diagnostic and remedial tools, and the theoretical frameworks they rely on are also quite diverse. A first attempt made for identifying these [START_REF] Itd | Theoretical frameworks of reference[END_REF] showed the existence of at least eight main theoretical frameworks: theory of didactical situations, anthropological theory of didactics, activity theory, instrumental approach, theory of semiotic mediation, social semiotics, socio-constructivism and constructionism, not to mention the theoretical approaches referred to in the AIED community and mobilized in the design of digital artefacts (Grandbastien & Labat, 2006).

For facilitating research collaboration, TELMA teams decided first to structure their collaborative work regarding the design and use of digital technologies around two main issues: representations and contexts, and to produce a description of each team according to common categories: main research aims, theoretical frameworks of references, digital tools designed and used… in order to make visible similarities and differences. As mentioned above, the descriptions produced evidenced a striking diversity in terms of theoretical frameworks, language and concepts used, and the difficulty we had to understand up to what point and how these differences affected our respective research and perspectives on the issues at stake. The notion of didactical functionality (see below) was then introduced as a reading key, general enough and based on elements relevant for all the teams, to be used to describe and compare frameworks. It was also decided to ask each team to select some few publications it considered the most appropriate for promoting mutual understanding and to work on these. Soon enough we experienced the limitation of such an enterprise: the reading of selected papers gave us only a rather superficial view of the exact role played by theoretical frames in our respective research projects.

Theoretical frames were of course evoked or even discussed but their links with the details of the actual research work were missing or remained fuzzy. The idea of developing a specific methodology: the cross-experimentation methodology, presented in the next part, emerged from the awareness of these limitations.

TELMA CONSTRUCTS

The first construct introduced in TELMA was the notion of didactical functionality. It was seen as a reading key as mentioned above and a means to link theoretical reflection and practice, helping us approach theories in more operational terms, beyond the declarative level dominating in the set of selected papers.

The notion of didactical functionality

The notion of didactical functionality (Cerulli et al, 2005) indeed individuates three different dimensions to be taken into account when considering a learning environment integrating one or several digital artefacts, for purpose of design or analysis of use:

• a set of features/characteristics of the considered digital artefact(s); • one (or a few coordinated) educational goal(s); • the modalities of use of the artefact(s) in the teaching and learning activity enacted to reach such goal(s).

These three dimensions are not independent of course: although characteristics and features of a digital tool can be identified through an a priori inspection, these features only become functionally meaningful when understood in relation to the educational goal for which the artefact is being used in a given context and to the modalities of its use. Nevertheless, identifying and distinguishing these dimensions helped us structure the reflection and analysis, and approach theoretical frameworks in operational terms. For progressing in the understanding of our similarities and differences, we needed then to complement this structure by appropriate descriptors or categories. This was the source of the notion of key concern we introduce below.

The notion of key concern

In spite of its limitations, the analysis of selected papers carried out showed that the different teams shared evident common sensitivities (for instance common sensitivity to semiotic and instrumental issues, to the social and situated dimensions of learning 1-4 Joint Meeting PME 32 -NA XXX -2008 processes), but they generally took these into consideration through different constructs and approaches. Retrospectively, the existence of such common sensitivities has nothing strange: even if we live in different educational cultures and have different trajectories, we are partly facing similar challenges and issues. Seeing theoretical frameworks and constructs as tools that we build for understanding and addressing challenges and issues, we thus conjectured that, for comparing and identifying possible productive connections between our respective theoretical frameworks and concepts, a good strategy could be to approach theories and concepts through the main sensitivities and needs they try to respond to. For tracing these common sensitivities and needs, we needed a common language not dependent on some particular theoretical approach. This was the source of the notion of key concern. A set of key concerns was thus attached to each dimension of the notion of didactical functionality, expressing the main sensitivities evidenced by the analysis carried out in the first phase of TELMA work [START_REF] Artigue | Towards a methodological tool for comparing the use of learning theories in technology enhanced learning in mathematics[END_REF].

If we consider for instance, the first dimension of the notion of didactical functionality corresponding to the analysis of the tool for identifying potentially interesting characteristics, we distinguished between different dimensions, questioning the usability of the tool, how the mathematical knowledge of the domain is implemented in the tool and what kind of relationships with mathematical objects this implementation allows, the forms of social and didactic interactions offered by the tool, the distance with institutional and cultural objects. This resulted in a set of 8 different key concerns for this dimension.

The theoretical frame(s) that a team relies on contribute to creating a partial hierarchy between key concerns. We decided to use these hierarchies, once identified, for organizing the comparison and connection between theoretical frameworks that we wanted to achieve, considering that priority had to be given to the cases where the same key concern or set of key concerns was given a high position by two or more different teams. In such cases, we expected to be able to trace how similar or close needs were fulfilled by different theoretical constructions, better understand the functionality of these, and infer from that possible interesting connections.

We had thus a structure and the meta-language of concerns for approaching theoretical connection, but what made these tools productive was the crossexperimentation methodology we developed for supporting the analysis.

The cross-experimentation methodology

The cross-experimentation methodology was supposed to enable comparison among teams highlighting similarities and differences in their research approaches. In order to do this TELMA teams developed a set of simultaneous teaching experiments according to the principles described below.

First of all it was decided that each team would develop a teaching experiment making use of an IT-based tool developed by another team. This was expected to induce deeper exchanges between the teams, and to make more visible the influence of theoretical frames through comparison of the vision of didactical functionalities developed by the designers of the digital artefacts and by the teams using these in the cross-experimentation. These simultaneous experiments needed to be gathered together to allows comparisons. For this reason it was decided the collaborative development of a common set of guidelines expressing questions to be addressed by each designing and experimenting team in order to frame the process of cross-team communication. This document was meant to draw a framework of common questions providing a methodological tool for comparing the theoretical basis of the individual studies, their methodologies and outcomes. Furthermore, to increase the visibility of theoretical choices and discussions, and also to make the experimental situation more realistic, it was decided that in each team PHD students and young researchers would be in charge of the experimentation.

Finally the range of some variables was limited: in order to facilitate the comparison between the different experimental settings, it was agreed to address common mathematical knowledge domains (fractions and introduction to algebra), to carry out the experiments with students between the 5th to 8th grade, and to perform classroom experiments of about the same duration (one month).

These principles were put in practice through an on-line collaborative activity that brought the involved young researchers characterised by the 4 main phases: 1. Production of a pre-classroom experiment version of the guidelines, containing plans for each experiment and answers to some questions (a priori questions); 2. Implementation of the classroom experiments; 3. Analysis of the experiments; 4. Production of the final version of the guidelines containing answers to all of the addressed questions (including the a posteriori questions).

Each phase was interlaced with reflection tasks were the involved researchers were requested to review in-itinere the other teams' answers to the questions contained in the guidelines, and to comment on them and ask for clarifications. In this way a constant dialogue could be set up, enabling researchers to bring to light implicit assumptions and to compare the different teams' approaches (Cerulli & al, 2007). In a sense the guidelines may be considered both as a product and as a tool supporting TELMA collaborative work. A product in the sense that the final version contains questions and answers to questions as well as plans, descriptions of the experiments and results. A tool in the sense that the guidelines structured each team's work by:

• providing research questions concerning contexts, representations, and theoretical frameworks;

• establishing the time when to address each question (ex. before, or after the classroom experiment, etc.); • establishing common concerns to focus on when describing classroom experiments, on the basis of the definition of DF; • gathering, under the same document, the answers provided by each team to the chosen questions, in a format that could possibly help comparisons.

The guidelines were finally complemented by a final analysis of the cross experiment based on a set of interviews: a senior researcher in each team, who was not directly involved with the experimental work, interviewed the young researchers who carried out the field experiments [START_REF] Artigue | Comparison of theories in technology enhanced learning in mathematics[END_REF]. Interviews followed a specific technique named "interview for explicitation" (Vermesch & Maurel, 1997): young researchers were asked to tell what they had done and how, but they were not directly questioned about the rationale for their actions.

THE LESSONS DRAWN FROM THE TELMA CROSS-EXPERIMENT

As was expected, the cross-experiment methodology, thanks to the perturbation it introduced in the normal functioning of the research teams, contributed to make visible the invisible, explicit the implicit. The space limitations of this research report do not allow us to enter into the necessary details, but we will try to show some important lessons that we drew from this cross-experimentation regarding both the role played by theoretical frames in design and analysis, and the needs and potentials in terms of coordination of theoretical frames. In the oral presentation, we plan to illustrate these results by using the two particular cases which are provided by the TELMA teams of the two co-authors of this research report: the DIDIREM team which experimented a digital artefact: Arilab, designed by the ITD team and the ITD team which experimented a digital artefact: Aplusix, designed by the Metah French team sharing the same didactical culture as DIDIREM.

The cross-experiment confirmed the conjectured relationship between theoretical frames and the key concern hierarchy, and showed the precise effects of this relationship in the design of the experiments, from the selection of the digital artefact to be experimented, the type of tasks proposed to the students, the diversity of semiotic mediations considered and the role given to these, the granularity in the planning of their management, the respective role given to the teacher and the student, to the attention paid to the distance with institutional and cultural habits. Moreover, it was evidenced that this influence was more or less conscious to the researchers. Familiar constructs were often used in a naturalized way and that was also the case regarding values. For that reason, the reflective interviews introduced in the cross-experimentation methodology were especially productive.

Another important result was that, even if important, the role of theoretical frames and concerns in shaping the design was limited. Answers to the guideline questionnaires and interviews evidenced the existing gap between what the theories offered and the decisions to be taken in the design. A lot of design decisions were determined by usual habits and experience and not under the control of theory. The same occurred in the implementation of the experimental design. Moreover, it clearly appeared that, for a given team, the hierarchy of key concerns was dependant on the moment of the experimentation: for instance concerns which played major role in the design of the experiment were less apparent in the analysis of the experiment. Vice versa, during the analysis phase, researchers often realized that they had underestimated specific needs in the design, and this awareness also contributed to move the concern hierarchy. They also faced unexpected events that were not so unexpected when adopting other theoretical perspectives, for instance those offered by other teams.

More generally, regarding connection and integration issues between theoretical frames, we draw from this experience a number of lessons potentially helpful for future research. We list below three of these.

The necessity of distinguishing, when looking at integration, possibilities and needs between design and a posteriori analysis. The economical and coherence needs of design are different of those of a posteriori analysis. Incorporating two many different theoretical frames can make design quite impossible, but in a posteriori analysis introducing new theoretical frames for instance for explaining unexpected events, producing alternative explanations, is easier and can be an effective support towards theoretical integration. For instance, the cross-experiment made clear that the theory of didactic situations and theory of semiotic mediation, which have a crucial role in design for the DIDIREM and the ITD team respectively, induce to control and anticipate in the design of an experiment is quite different but that each vision has its own coherence and leads the design in a different and potentially productive direction. But we also got the evidence that the theoretical tools of one approach can enrich the a posteriori analysis of the other one.

The fact that the hierarchy of concerns can be exploited for looking at possible theoretical connections in different ways. In TELMA work, similarities in hierarchies were first exploited for establishing connections between theoretical frames and concepts, but contrasted priorities can also been exploited for looking at possible complementarities between theoretical frames.

The fact that progressing in the comparison and connection between theoretical frames needs the development of specific structures and languages making the communication possible. In our case, these structure and languages were provided by the notion of didactical functionality and the language of concerns. They obliged us

1-8

Joint Meeting PME 32 -NA XXX -2008 to approach theories in terms of functionalities and this approach was really productive.

Beyond that, progression needs also the building of some form of collaborative practice supporting the comparison and connection work. Knowledge in this domain as in others cannot only result from readings, explanations and discussions. In our case, the cross-experimentation was asked to play this role, and the results it allowed us to achieve led us to reinvest this methodology in a new and more ambitious European project: the Remath project (Representing Mathematics with Digital Technologies) where the collaboration is extended towards the development of digital artefacts, of a common language for scenarios, and of an integrative platform MathDils. In this project, each team experiments both faliliar and alien digital artefacts in realistic contexts and cross-experiments. Moreover each team experiments both its own ILE and an alien ILE in realistic contexts, and the methodological tools built in TELMA are no longer only used to foster communication per se but also to achieve specific common research goals.


Toby Dylan

Hocking -Toby Hocking

Clusterpath An Algorithm for Clustering using Convex Fusion Penalties

We present a new clustering algorithm by proposing a convex relaxation of hierarchical clustering, which results in a family of objective functions with a natural geometric interpretation. We give efficient algorithms for calculating the continuous regularization path of solutions, and discuss relative advantages of the parameters. Our method experimentally gives state-of-the-art results similar to spectral clustering for non-convex clusters, and has the added benefit of learning a tree structure from the data.

Introduction

In the analysis of multivariate data, cluster analysis is a family of unsupervised learning techniques that allows identification of homogenous subsets of data. Algorithms such as k-means, Gaussian mixture models, hierarchical clustering, and spectral clustering allow recognition of a variety of cluster shapes. However, all of these methods suffer from instabilities, either because they are cast as non-convex optimization problems, or because they rely on hard thresholding of distances. Several convex clustering methods have been proposed, but some only focus on the 2-class problem [START_REF] Xu | Maximum margin clustering[END_REF], and others require arbitrary fixing of minimal cluster sizes in advance [START_REF] Bach | DIFFRAC: a discriminative and flexible framework for clustering[END_REF]. The main contribution of this work is the development of a new convex hierarchical clustering algorithm that attempts to address these concerns.

In recent years, sparsity-inducing norms have emerged as flexible tools that allow variable selection in penalized linear models. The Lasso and group Lasso are now well-known models that enforce sparsity or group-wise sparsity in the estimated coefficients [START_REF] Tibshirani | Regression Shrinkage and Selection Via the Lasso[END_REF][START_REF] Yuan | Model selection and estimation in regression with grouped variables[END_REF]. Another example, more useful for clustering, is the fused Lasso signal approximator (FLSA), which has been used for segmentation and image denoising [START_REF] Tibshirani | Sparsity and smoothness via the fused lasso[END_REF]. Furthermore, several recent papers have proposed optimization algorithms for linear models using ℓ 1 [CKL + 10, SH10] and ℓ 2 [START_REF] Vert | Fast detection of multiple change-points shared by many signals using group LARS[END_REF] fusion penalties. This paper extends this line of work by developing a family of fusion penalties that results in the "clusterpath," a hierarchical regularization path which is useful for clustering problems.

Motivation by relaxing hierarchical clustering

Hierarchical or agglomerative clustering is calculated using a greedy algorithm, which for n points in R p recursively joins the points which are closest together until all points are joined. For the data matrix X ∈ R n×p this suggests the optimization problem

min α∈R n×p 1 2 ||α -X|| 2 F subject to i<j 1 αi =αj ≤ t, (1)
where ||•|| 2 F is the squared Frobenius norm, α i ∈ R p is row i of α, and 1 αi =αj is 1 if α i = α j , and 0 otherwise. We use the notation i<j = n-1 i=1 n j=i+1 to sum over all the n(n-1)/2 pairs of data points. Note that when we fix t ≥ n(n-1)/2 the problem is unconstrained and the solutions are α i = X i for all i. If t = n(n-1)/2-1, we force one pair of coefficients to fuse, and this is equivalent to the first step in hierarchical clustering. Furthermore, when t = 0, the solutions are clearly α i = X = n i=1 X i /n. In general this is a difficult combinatorial optimization problem.

Instead, we propose a convex relaxation, which results in the family of optimization problems defined by

min α∈R n×p 1 2 ||α -X|| 2 F subject to Ω q (α) = i<j w ij ||α i -α j || q ≤ t, (2)
where w ij > 0, and || • || q , q ∈ {1, 2, ∞} is the ℓ q -norm on R p , which will induce sparsity in the differences of the rows of α. When rows fuse we say they form a cluster, and the continuous regularization path of optimal solutions formed by varying t is what we call the "clusterpath." This parameterization in terms of t is cumbersome when comparing datasets since we take 0 ≤ t ≤ Ω q (X), so we introduce the following parametrization with 0 ≤ s ≤ 1:

min α∈R n×p 1 2 ||α -X|| 2 F subject to Ω q (α)/Ω q (X) ≤ s. (3)
The equivalent Langrangian dual formulation will also be convenient for optimization algorithms:

min α∈R n×p f q (α, X) = 1 2 ||α -X|| 2 F + λΩ q (α). (4)
Equivalent parameter values and their solutions for the ends of the path are compared in Table 1.

Table 1: Parameters and solutions for the ends of the clusterpath. Note that in general, there is no closed form expression for λ max , and

X = n i=1 X i /n. t s λ α * i Ω(X) 1 0 Xi 0 0 λmax X
The above optimization problems require the choice of predefined, pair-specific weights w ij > 0, which can be used to control the geometry of the solution path. In most of our experiments we use weights that decay with the distance between points w ij = exp(-γ||X i -X j || 2 2 ), which results in a clusterpath that is sensitive to local density in the data. Another choice for the weights is w ij = 1, which allows efficient computation of the ℓ 1 clusterpath ( §2.2). Choosing weights based on some supplemenary data space w ij = exp(-γ||Y i -Y j || 2 ) could also be interesting. For example, for clustering pixels of an image into objects, we could take X ∈ R n×2 to be the matrix of pixel positions, and Y ∈ R n×p to be the matrix of visual features for each pixel.

Visualizing the geometry of the clusterpath

This optimization problem has an equivalent geometric interpretation (Figure 1). For the identity weights w ij = 1, the solution corresponds to the closest points α to the points X, subject to a constraint on the sum of distances between pairs of points. For general weights, we constrain the total area of the rectangles of width w ij between pairs of points.

In this work we develop dedicated algorithms for solving the clusterpath which allow scaling to large data, but initially we used cvxmod for small problems [START_REF] Mattingley | CVXMOD: Convex optimization software in Python (web page and software[END_REF], as the authors do in a similar independent formulation [START_REF] Lindsten | Clustering using sum-of-norms regularization; with application to particle filter output computation[END_REF].

We used cvxmod to compare the geometry of the clusterpath for several choices of norms and weights (Figure 2). Note the piecewise linearity of the ℓ 1 and ℓ ∞ clusterpath, which can be exploited to find the solutions using efficient path-following homotopy algorithms. Furthermore, it is evident that the ℓ 2 path is invariant to rotation of the input data X, whereas the others are not.

Identity weights, t = Ω(X)

ℓ 2 ℓ 2 ℓ 2 ℓ 1 ℓ 1 ℓ 1 ℓ 1 ℓ 1 ℓ 1 ℓ ∞ ℓ ∞ ℓ ∞ X 1 X 2 X 3
Decreasing weights, t = Ω(X)

w 12 w 13 w 23 X 1 X 2 X 3
Decreasing weights after join, t < Ω(X)

w 12 w 13 X 1 X 2 X 3 α 1 α C = α 2 = α 3
Figure 1: Geometric interpretation of the optimization problem (2) for data X ∈ R 3×2 . Left: with the identity weights w ij = 1, the constraint Ω q (α) = i<j w ij ||α i -α j || q ≤ t is the ℓ q distance between all pairs of points, shown as grey lines. Middle: with general weights w ij , the ℓ 2 constraint is the total area of rectangles between pairs of points. Right: after constraining the solution, α 2 and α 3 fuse to form the cluster C, and the weights are additive: w 1C = w 12 + w 13 .

Optimization

2.1 A homotopy algorithm for the ℓ 1 solutions

For the problem involving the ℓ 1 penalty, we first note that the problem is separable on dimensions. The cost function can be written as

f 1 (α, X) = 1 2 ||α -X|| 2 F + λΩ 1 (α) = 1 2 n i=1 p k=1 (α ik -X ik ) 2 + λ i<j w ij p k=1 |α ik -α jk | = p k=1   1 2 n i=1 (α ik -X ik ) 2 + λ i<j w ij |α ik -α jk |   = p k=1 f 1 (α k , X k ),
where α k ∈ R n is the k-th column from α. Thus, solving the minimization with respect to the entire matrix X just amounts to solving p separate minimization subproblems:

min α∈R n×p f 1 (α, X) = p k=1 min α k ∈R n f 1 (α k , X k ).
For each of these subproblems, we can exploit the FLSA path algorithm [START_REF] Hoefling | A path algorithm for the fused lasso signal approximator[END_REF]. This is a homotopy algorithm similar to the LARS that exploits the piecewise linearity of the path to very quickly calculate the entire set of solutions [START_REF] Efron | Least angle regression[END_REF].

norm = 1 X X norm = 2 X X norm = ∞ X X γ = 0 γ = 1
Figure 2: Some random normal data X ∈ R 10×2 were generated (white dots) and their mean X is marked in the center. The clusterpath (black lines) was solved using cvxmod for 3 norms (panels from left to right) and 2 weights (panels from top to bottom), which were calculated using w ij = exp(-γ||X i -X j || 2 ). For γ = 0, we have w ij = 1.

In the LARS, variables jump in and out the active set, and we must check for these events at each step in the path. The analog in the FLSA path algorithm is the necessity to check for cluster splits, which occur when the optimal solution path requires unfusing a pair coefficients. Cluster splits were not often observed on our experiments, but are also possible for the ℓ 2 clusterpath, as illustrated in Figure 4. The FLSA path algorithm checks for a split of a cluster of size n C by solving a max-flow problem using a push-relabel algorithm, which has complexity O(n 3 C ) [START_REF] Cormen | Introduction to Algorithms[END_REF]. For large data sets, this can be prohibitive, and for any clustering algorithm, splits make little sense.

One way around this bottleneck is to choose weights w in a way such that no cluster splits are possible in the path. The modified algorithm then only considers cluster joins, and results in a complexity of O(n log n) for a single dimension, or O(pn log n) for p dimensions. One choice of weights that results in no cluster splits is the identity weights w ij = 1, which we prove below.

2.2

The ℓ 1 clusterpath using w ij = 1 contains no splits

The proof will establish a contradiction by examining the necessary conditions on the optimal solutions during a cluster split. We will need the following lemma.

Lemma 1. Let C = {i : α i = α C } ⊆ {1, .
.., n} be the cluster formed after the fusion of all points in C. At any point in the regularization path, the slope of its coefficient is given by

v C = dα C dλ = 1 |C| j ∈C w Cj sign(α j -α C ). (5)
Proof. Consider the following sufficient optimality condition, for all i = 1, . . . , n: [START_REF] Hoefling | A path algorithm for the fused lasso signal approximator[END_REF]. We can rewrite the optimality condition for all i ∈ C:

0 = α i -X i + λ j =i α i =α j w ij sign(α i -α j ) + λ j =i α i =α j w ij β ij , with |β ij | ≤ 1 and β ij = -β ji
0 = α C -X i + λ j ∈C w ij sign(α C -α j ) + λ i =j∈C w ij β ij .
Furthermore, by summing each of these equations, we obtain the following:

α C = XC + λ |C| j ∈C w Cj sign(α j -α C ),
where XC = i∈C X i /|C| and w Cj = i∈C w ij . Taking the derivative with respect to λ gives us the slope v C of the coefficient line for cluster C, proving Lemma 1.

We will use Lemma 1 to prove by contradiction that cluster splitting is impossible for the case w ij = 1 for all i and j.

Theorem 1. Taking w ij = 1 for all i and j is sufficient to ensure that the ℓ 1 clusterpath contains no splits.

Proof. Consider at some λ the optimal solution α, and let C be a cluster of any size among these optimal solutions. Denote the set C = {i : α i > α C } the set of indices of all larger optimal coefficients and C = {i : α i < α C } the set of indices of all smaller optimal coefficients. Note that C ∪ C ∪ C = {1, . . . , n}. Now, assume C splits into C 1 and C 2 such that α 1 > α 2 . By Lemma 1, if this situation constitutes an optimal solution, then the slopes are:

v C1 = 1 |C 1 |   j∈C w jC1 - j∈C2 w jC1 - j∈C w jC1   v C2 = 1 |C 2 |   j∈C w jC2 + j∈C1 w jC2 - j∈C w jC2   .
For the identity weights, this simplifies to

v C1 = |C| -|C 2 | -|C| v C2 = |C| + |C 1 | -|C|.
Thus v C1 < v C2 which contradicts the assumption that α 1 > α 2 , forcing us to conclude that no split is possible for the identity weights.

Thus the simple FLSA algorithm of complexity O(n log n) without split checks is sufficient to calculate the ℓ 1 clusterpath for the identity weights, as shown in Figure 3.

Furthermore, since the clusterpath is strictly agglomerative on each dimension, it is also strictly agglomerative when independently applied to each column of a matrix of data. Thus the ℓ 1 clusterpath for a matrix of data is strictly agglomerative, and results in an algorithm of complexity O(pn log n). This is an interesting alternative to hierarchical clustering, which normally requires O(pn 2 ) space and time for p > 1. The ℓ 1 clusterpath can be used when n is very large, and hierarchical clustering is not feasible.

The proposed homotopy algorithm only gives solutions to the ℓ 1 clusterpath for identity weights, but since the ℓ 1 clusterpath in 1 dimension is a special case of the ℓ 2 clusterpath, the algorithms proposed in the next subsection also apply to solving the ℓ 1 clusterpath with general weights.

An active-set descent algorithm for the ℓ 2 solutions

For the ℓ 2 problem, we have the following cost function:

f 2 (α, X) = 1 2 ||α -X|| 2 F + λΩ 2 (α),
A subgradient condition sufficient for an optimal α is for all i ∈ 1, ..., n:

0 = α i -X i + λ j =i α j =α i w ij α i -α j ||α i -α j || 2 + λ j =i α j =α i w ij β ij , with β ij ∈ R p , ||β ij || 2 ≤ 1 and β ij = -β ji .
Summing over all i ∈ C gives the subgradient for the cluster C:

G C = α C -XC + λ |C| j ∈C w Cj α C -α j ||α C -α j || 2 , (6)
where XC = i∈C X i /|C| and w Cj = i∈C w ij .

To solve the ℓ 2 clusterpath, we propose a subgradient descent algorithm, with modifications to detect cluster fusion and splitting events (Algorithm 1). Note that due to the continuity of the ℓ 2 clusterpath, it is advantageous to use warm restarts between successive calls to SOLVE-L2, which we do using the values of α and clusters. Figure 3: The ℓ 1 clusterpath calculated using the homotopy algorithm (lines) and cvxmod (points) for an X ∈ R 5 and w ij = 1.

Algorithm 1 CLUSTERPATH-L2 Input: data X ∈ R n×p , weights w ij > 0, starting λ > 0 α ← X clusters ← {{1}, ..., {n}} while | clusters | > 1 do α, clusters ← SOLVE-L2(α, clusters, X, w, λ) λ ← λ × 1.5
if we are considering cluster splits then clusters ← {{1}, ..., {n}} end if end while return table of all optimal α and λ values. Surprisingly, the ℓ 2 path is not always agglomerative, and in this case to reach the optimal solution requires restarting clusters = {{1}, ..., {n}}. The clusters will rejoin in the next call to SOLVE-L2 if necessary. This takes more time but ensures that the optimal solution is found, even if there are splits in the clusterpath, as in Figure 4.

We conjecture that there exist certain choices of w for which there are no splits in the ℓ 2 clusterpath. However, a theorem analogous to Theorem 1 that establishes necessary and sufficient conditions on w and X for splits in the ℓ 2 clusterpath is beyond the scope of this article. We have not observed cluster splits in our calculations of the path for identity weights w ij = 1 and decreasing weights w ij = exp(-γ||X i -X j || 2

2 ), and we conjecture that these weights are sufficient to ensure no splits.

SUBGRADIENT-L2 calculates the subgradient from (6), for every cluster C ∈ clusters.

We developed 2 approaches to implement SUBGRADIENT-STEP. In both cases we use the update α ← α-rG. With decreasing step size r = 1/ iteration, the algorithm takes many steps before converging to the optimal solution, even though we restart the iteration count after cluster fusions. The second approach we used is a line search. We evaluated the cost function at several points r and picked the r with the lowest cost. In practice, we observed fastest performance when we alternated every other step between decreasing and line search.

DETECT-CLUSTER-FUSION calculates pairwise differences between points and checks for cluster fusions, returning the updated matrix of points α and the new list of clusters. When 2 clusters C 1 and C 2 fuse to produce

Algorithm 2 SOLVE-L2

Input: initial guess α, initial clusters, data X, weights w,

regularization λ G ← SUBGRADIENT-L2(•) while ||G|| 2 F > ǫ opt do α ← SUBGRADIENT-STEP(•) α, clusters ← DETECT-CLUSTER-FUSION(•) G ← SUBGRADIENT-L2(•)
end while return α, clusters a new cluster C, the coefficient of the new cluster is calculated using the weighted mean:

α C = |C 1 |α C1 + |C 2 |α C2 |C 1 | + |C 2 | . ( 7
)
We developed 2 methods to detect cluster fusions. First, we can simply use a small threshhold on ||α C1 -α C2 || 2 , which we usually take to be some fraction of the smallest nonzero difference in the original points

||X i -X j || 2 .
Second, to confirm that the algorithm does not fuse points too soon, for each possible fusion, we checked if the cost function decreases. This is similar to the approach used by [START_REF] Friedman | Pathwise coordinate optimization[END_REF], who use a coordinate descent algorithm to optimize a cost function with an ℓ 1 fusion penalty. Although this method ensures that we reach the correct solution, it is quite slow since it requires evaluation of the cost function for every possible fusion event.

The Frank-Wolfe algorithm for ℓ ∞ solutions

We consider the following ℓ ∞ problem:

min α∈R n×p f ∞ (α, X) = 1 2 ||α -X|| 2 F + λΩ ∞ (α). (8)
This problem has a piecewise linear regularization path which we can solve using a homotopy algorithm to exactly calculate all the breakpoints [RZ07, ZRY09]. However, empirically, the number of breakpoints in the path grows fast with p and n, leading to instability in the homotopy algorithm. Instead, we show show that our problem is equivalent to a norm minimization over a polytope, for which an efficient algorithm exists [START_REF] Frank | An algorithm for quadratic programming[END_REF].

Using the dual formulation of the ℓ ∞ norm, the regularization term is equal to:

Ω ∞ (α) = i<j w ij max s ij ∈R p ||s ij || 1 ≤1 s T ij (α i -α j ).
Denoting by r i = j>i s ij w ij -j<i s ji w ij ∈ R p , and by R the set of constraints over R = (r 1 , . . . , r n ) such that the constraints over s ij are respected, we have:

Ω ∞ (α) = max R∈R tr R T α .
Since R is defined as a set of linear combinations of ℓ 1 -ball inequalities, R is a polytope. Denoting by Z = X-λR and

Z = {Z | 1 λ (X -Z) ∈ R}, it is straightforward to prove that problem (8) is equivalent to: min α∈R n×p max Z∈Z H(α, Z) = α -Z 2 F -Z 2 F ,
where strong duality holds [START_REF] Boyd | Convex Optimization[END_REF]. For a given Z, the minimum of H in α is obtained by α = Z, leading to a norm minimization over the polytope Z.

min Z∈Z 1 2 Z 2 F . (9)
This problem can be solved efficiently by using the Frank-Wolfe algorithm [START_REF] Frank | An algorithm for quadratic programming[END_REF]. This algorithm to minimize a quadratic function over a polytope may be used as soon as it is possible to minimize linear functions in closed form. It is also known as the minimum-norm-point algorithm when applied to submodular function minimization [START_REF] Fujishige | The minimum-norm-point algorithm applied to submodular function minimization and linear programming[END_REF]. In practice, it is several orders of magnitude faster than other common discrete optimization algorithms, but there is no theoretical guarantee on its complexity [START_REF] Krause | Beyond convexity: Submodularity in machine learning[END_REF].

The spectral clusterpath: a completely convex formulation of spectral clustering

For spectral clustering, the usual formulation uses eigenvectors of the normalized Laplacian as the inputs to a standard clustering algorithm like k-means [START_REF] Ng | On spectral clustering: Analysis and an algorithm[END_REF]. Specifically, for several values of γ, we compute a pairwise affinity matrix W such that W ij = exp(-γ||X i -X j || 2 2 ) and a Laplacian matrix L = D -W where D is a diagonal matrix such that D ii = n j=1 W ij . For each value of γ, we run k-means on the normalized eigenvectors associated with k smallest eigenvalues of L, then keep the γ with lowest reconstruction error.

Some instability in spectral clustering may come from the following 2 steps. First, the matrix of eigenvectors is formed by hard-thresholding the eigenvalues, which is unstable when several eigenvalues are close. Second, the clusters are located using the k-means algorithm, which attempts to minimize a non-convex objective. To relax these potential sources of instability, we propose the "spectral clusterpath," which replaces (a) hard-thresholding by soft-thresholding and (b) k-means by the clusterpath.

Concretely, we call (Λ i ) 1≤i≤n the nontrivial eigenvalues sorted in ascending order, and we write the matrix of transformed eigenvectors to cluster as V E, where V is the full matrix of sorted nontrivial eigenvectors and E is the diagonal matrix such that E ii = e(Λ i ), and e : R → R ranks importance of eigenvectors based on their eigenvalues. Standard spectral clustering takes e 01 (x) = 1 x≤Λ k such that only the first k eigenvalues are selected. This is a non-convex hard-thresholding of the full matrix of eigenvectors. We propose the exponential function e exp (x) = exp (-νx), with ν > 0, as a convex relaxation. This smooth decreasing function continuously scales the eigenvectors based on the eigenvalues.

Results

Our model poses 3 free parameters to choose for each matrix to cluster: norm, weights, and regularization. On one hand, this offers the flexibility to tailor the geometry of the solution path and number of clusters for each data set. On the other hand, this poses model selection problems as training clustering models is not straightforward. Many heuristics have been proposed for automatically choosing the number of clusters [START_REF] Tibshirani | Estimating the number of clusters in a data set via the gap statistic[END_REF], but it is not clear which of these is applicable to any given data set.

In the experiments that follow, we chose the model based on the desired geometry of the solution path and number of clusters. We generally expect rotation invariance in multivariate clustering models, so we chose the ℓ 2 norm with Gaussian weights to encourage sensitivity to local density.

Verification on non-convex half-moon clusters

To compare our algorithm to other popular methods in the setting of non-convex clusters, we generated data in the form of 2 interlocking half-moons (Figure 5), which we used as input for several clustering algorithms (Table 2). We used the original data as input for k-means, Gaussian mixtures, average linkage hierarchical clustering, and the ℓ 2 clusterpath with γ = 2. For the other methods, we use the eigenvectors from spectral clustering as input. Each algorithm uses 2 clusters and performance is measured using the normalized Rand index, which varies from 1 for a perfect match to 0 for completely random assignment [START_REF] Hubert | Comparing partitions[END_REF].

In the original input space, hierarchical clustering and k-means fail, but the clusterpath is able to identify the clusters as well as the spectral methods, and has the added benefit of learning a tree from the data. However, the clusterpath takes 3-10 times more time than the spectral methods. Of the methods that cluster the eigenvectors, the most accurate 2 methods use e exp rather than e 01 , providing evidence that the convex relaxation stabilizes the clustering.

Recovery of many Gaussian clusters

We also tested our algorithm in the context of 25 Gaussian clusters arranged in a 5×5 grid in 2 dimensions (Figure 6). 20 data points were generated from each cluster, and the resulting data were clustered using k-means, hierarchical clustering, and the weighted ℓ 2 clusterpath. The clusterpath performs similarly to hierarchical clustering, which exactly recovers the clusters, and k-means fails. Thus, the clusterpath may be useful for clustering tasks that involve many clusters.

Conclusions

We proposed a framework for clustering using linear models and several convex pairwise fusion penalties, which lead to efficient algorithms for optimization (Table 4). The ℓ 1 path-following homotopy algorithm easily scales to thousands of points. The other proposed algorithms can be directly applied to hundreds of points, and could be applied to larger datasets by, for example, adding a preprocessing step using k-means. The algorithms were implemented in R, C++, and MATLAB, and will be published soon.

Our experiments demonstrated the flexibility of the ℓ 2 clusterpath for the unsupervised learning of non-convex clusters, large numbers of clusters, and hierarchical structures. We also observed that relaxing hard-thresholding in spectral clustering is useful for increasing clustering accuracy and stability. For the iris data, the clusterpath performed as well as hierarchical clustering, and is more stable than k-means.

We proved that the identity weights are sufficient for the ℓ 1 clusterpath to be strictly agglomerative. Establishing necessary and sufficient conditions on the weights for the ℓ 2 problem is an avenue for further research.

To extend these results, we are currently pursuing research into optimizing a linear model with a non-identity design matrix and the clusterpath penalty. We note that there could be a future application for the algorithms presented in this article in solving the proximal operator, which is the same as (4) for the clusterpath penalty.

Figure 4 :

 4 Figure 4: An example of a split in the ℓ 2 clusterpath for X ∈ R 4×2 . Data points are labeled with numbers, the CLUSTERPATH-L2 is shown as lines, and solutions from cvxmod are shown as circles. w 12 = 9, w 13 = w 24 = 20, and w ij = 1 for the others (best seen in color).

Figure 5 :

 5 Figure 5: Typical results for 5 clustering algorithms applied to 2 half-moon non-convex clusters. The ℓ 2 clusterpath tree learned from the data is also shown. Spectral clustering and the clusterpath correctly identify the clusters, while average linkage hierarchical clustering and k-means fail.

Figure 6 :

 6 Figure 6: A grid of gaussian clusters breaks k-means but not the clusterpath. The weighted ℓ 2 clusterpath is shown as grey lines.

Figure 8 :

 8 Figure 8: Performance on the iris and moons data, as measured by the normalized Rand index of models with 2-11 clusters. The weighted ℓ 2 clusterpath was calculated using 3 different Gaussian weight parameters γ, and we compare with Gaussian Mixture Models (GMM), Hierarchical Clustering (HC), and k-means.

Table 2 :

 2 Mean and standard deviation of performance and timing of several clustering methods on identifying 20 simulations of the half-moons in Figure 5. Ng et al. uses L = I -D -1/2 W D -1/2 rather than L = D -W as discussed in the text. Large Rand means good clustering.

	Clustering method	Rand	SD Seconds	SD
	eexp spectral clusterpath 0.9967 0.0073	8.4920 2.6474
	eexp spectral kmeans	0.9967 0.0073	3.1078 0.0848
	clusterpath	0.9580 0.1262 29.4738 2.3122
	e01 Ng et al. kmeans	0.9538 0.1911	7.3772 0.4222
	e01 spectral kmeans	0.9149 0.1978	3.2647 0.2129
	Gaussian mixture	0.4254 0.1334	0.0783 0.0051
	average linkage	0.4047 0.1341	0.0557 0.0009
	kmeans	0.2668 0.0466	0.0011 0.0002

Table 3 :

 3 Performance of several clustering methods on identifying a grid of Gaussian clusters. Means and standard deviations from 20 simulations are shown.

	Clustering method	Rand	SD
	kmeans	0.8365 0.0477
	clusterpath	0.9955 0.0135
	average linkage hierarchical 1.0000 0.0000

Table 4 :

 4 Algorithms proposed to solve the clusterpath.AwknowledgementsFB was supported by grant SIERRA-ERC-239993. TDH was supported by grant DIGITEO-BIOVIZ-2009-25D. JPV was supported by grants ANR-07-BLAN-0311-03 and ANR-09-BLAN-0051-04.

	Norm	Properties	Algorithm	Complexity	Problem sizes
	1	piecewise linear, separable	homotopy	O(pn log n)	large ≈ 10 5
	2	rotation invariant	active-set descent	O(n 2 p)	medium ≈ 10 3
	∞	piecewise linear	Frank-Wolfe	unknown	medium ≈ 10 3

Application to clustering the iris data

To evaluate the clusterpath on a nontrivial task, we applied it and other common clustering methods to the scaled iris data (Figure 7). We calculated a series of clusterings using each algorithm and measured performance of each using the normalized Rand index (Figure 8).

The iris data have 3 classes, of which 2 overlap, so the Gaussian Mixture Model is the only algorithm capable of accurately detecting these clusters when k = 3. These data suggest that the clusterpath is not suitable for detecting clusters with large overlap. However, performance is as good as hierarchical clustering, less variable than k-means, and more stable as the number of clusters increases.

Additionally, Figure 8 shows that the clusterpath classification accuracy on the moons data increases as we increase the weight parameter γ. The 4-dimensional iris data were scaled, and the ℓ 2 clusterpath was calculated using Gaussian weights with γ = 1, and plotted using grey lines. The mean of the data is shown as a black dot.


S Vakeroudis

M Yor

‡ D Holcman

The Mean First Rotation Time of a planar polymer

We estimate the mean first time, called the mean rotation time (MRT), for a planar random polymer to wind around a point. This polymer is modeled as a collection of n rods, each of them being parameterized by a Brownian angle. We are led to study the sum of i.i.d. imaginary exponentials with one dimensional Brownian motions as arguments. We find that the free end of the polymer satisfies a novel stochastic equation with a nonlinear time function. Finally, we obtain an asymptotic formula for the MRT, whose leading order term depends on √ n and, interestingly, depends weakly on the mean initial configuration. Our analytical results are confirmed by Brownian simulations.

Introduction

This paper focusses mainly on some properties of a planar polymer motion and in particular, on the mean time that a rotation is completed around a fixed point. This mean rotation time (MRT) provides a quantification for the transition between a free two dimensional Brownian motion and a restricted motion. To study the distribution of this time, we shall use a simplified model of a polymer made of a collection of n two dimensional connected rods with Brownian random angles (Figure 1). To estimate the mean rotation time, we shall fix one end of the polymer. We shall examine how the MRT depends on various parameters such as the diffusion constant, the number of rods or their common length. Using some approximations and numerical simulations, the mean time for the two polymer ends to meet was estimated in dimension three [START_REF] Wilemski | Diffusion-controlled intrachain reactions of polymers. I Theory[END_REF][START_REF] Pastor | Diffusion limited first contact of the ends of a polymer: Comparison of theory with simulation[END_REF].

Although the windings of a (planar) Brownian motion, that is the number of rotations around one point in dimension 2 or a line in dimension three and its asymptotic behavior have been studied quite extensively [START_REF] Spitzer | Some theorems concerning two-dimensional Brownian Motion[END_REF][START_REF] Pitman | Asymptotic Laws of planar Brownian Motion[END_REF][START_REF] Gall | Etude asymptotique des enlacements du mouvement brownien autour des droites de l'espace[END_REF][START_REF] Gall | Some properties of planar Brownian motion[END_REF][START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF], little seems to be known about the mean time for a rotation to be completed for the first time. For an Ornstein-Uhlenbeck process, the first time that it hits the boundary of a given cone was recently estimated [START_REF] Vakeroudis | On hitting times of the winding processes of planar Brownian motion and of Ornstein-Uhlenbeck processes, via Bougerol's identity[END_REF].

The paper is organized as follows: in section 2, we present the polymer model. In section 3, we study a sum of i.i.d. exponentials of one dimensional Brownian motions. Interestingly, using a Central Limit Theorem, we obtain a new stochastic equation for the limit process. This equation describes the motion of the free polymer end. In section 4, we obtain our main result which is an asymptotic formula for E [τ n ] the MRT when the polymer is made of n rods of equal lengths l 0 with the first end fixed at a distance L from the origin and the Brownian motion is characterized by its rotational diffusion constant D. We find that the MRT depends logarithmically on the mean initial configuration and for nl 0 >> L and n ≥ 3, the leading order term is given by: 1. for a general initial configuration:

E [τ n ] ≈ √ n 8D   2 ln   E n k=1 e i √ 2D θ k (0) √ n   + Q   ,
where (θ k (0), 1 ≤ k ≤ n) are the initial angles of the polymer and Q ≈ 9.56, 2. for an initially stretched polymer:

E [τ n ] ≈ √ n 8D (ln(n) + Q) ,
3. for an average over uniformly distributed initial angles:

E [τ n ] ≈ √ n 8D Q,
where Q ≈ 9.62.

We confirm our analytical results with generic Brownian simulations with normalized parameters . Finally, in section 5 we discuss some related open questions.

2 Stochastic modeling of a planar polymer

Various models are available to study polymers: the Rouse model consists of a collection of beads connected by springs, while more sophisticated models account for bending, torsion and specific mechanical properties [START_REF] Rouse | A Theory of the Linear Viscoelastic Properties of Dilute Solutions of Coiling Polymers[END_REF][START_REF] Szabo | First passage time approach to diffusion controlled reactions[END_REF][START_REF] Doi | The Theory of Polymer Dynamics[END_REF]. Here, we consider a very crude approximation where a planar polymer is modeled as a collection of n rigid rods, with equal fixed lengths l 0 and we denote their extremities by (X 0 , X 1 , X 2 , . . . , X n ) (Figure 1) in a framework with origin 0. We shall fix one of the polymer ends X 0 = (L, 0) (where L > 0) on the x-axis. The dynamics of the i-th rod is characterized by its angle θ i (t) with respect to the x-axis. The overall polymer dynamics is thus characterized by the angles (θ 1 (t), θ 2 (t), . . . , θ n (t), t ≥ 0). Due to the thermal collisions in the medium, each angle follows a Brownian motion.

Thus, with

= denoting equality in law,

(θ i (t), i ≤ n) (law) = √ 2D (B i (t), i ≤ n) that is : dθ i (t) = √ 2D dB i (t), i ≤ n,
where D is the rotational diffusion constant and (B 1 (t), . . . , B n (t), t ≥ 0) is an n-dimensional Brownian motion (BM). The position of each rod can now be obtained as: t) .

X 1 (t) = L + l 0 e iθ 1 (t) , X 2 (t) = X 1 (t) + l 0 e iθ 2 (t) , . . . , X n (t) = X n-1 (t) + l 0 e iθn(
(1) In particular, the moving end is given by:

X n (t) = L + l 0 n k=1 e iθ k (t) = L + l 0 n k=1 e i √ 2DB k (t) , (2)
which can be written as:

X n (t) = R n (t)e iϕn(t) . (3)
Thus ϕ n (t) accounts for the rotation of the polymer with respect to the origin 0 and R n is the distance to the origin. In order to compute the MRT, we shall study a sum of exponentials of Brownian motions, a topic which often leads to surprising computations [START_REF] Yor | Exponential Functionals of Brownian Motion and Related Processes[END_REF]. First, we scale the space and time variables as follows:

l = L l 0 and t = t 2D . (4)
Equation (2) becomes:

X n (t) = l + n k=1 e i Bk (t) , (5)
where B1 (t), . . . , Bn (t), t ≥ 0 is an n-dimensional Brownian motion (BM), and for k = 1, . . . , n, using the scaling property of Brownian motion, we have:

Bk (t) ≡ 1 √ 2D B k (t) (law) = B k t 2D = B k ( t).
Before describing our approach, we first discuss the mean initial configuration of the polymer. It is given by:

c n = E n k=1 e iθ k (0) , (6)
where the initial angles θ k (0) are such that the polymer has not already made a loop. After scaling, the mean initial configuration becomes:

c n = E n k=1 e i θk (0) , (7)
with

θk (0) = Bk (0). (8)
From now on, we use θ k instead of θk , B instead of B and t instead of t.

Any segment in the interior of the polymer can hit the angle 2π around the origin, but we will not consider this as a winding event, although we could and in that case, the MRT would be different. Rather, we shall only consider that, given an initial configuration c n , the MRT is defined as:

MRT ≡ E{τ n |c n } ≡ E [τ n ] , (9)
where

τ n ≡ inf{t > 0, |ϕ n (t)| = 2π}. (10)
Thus, an initial configuration is not winding when

|ϕ n (0)| < 2π. (11)
Then, we can define the winding event using a one dimensional variable only.

In general, winding is a rare event and we expect that the MRT will depend crucially on the length of the polymer which will be quite long. Interestingly, the rotation is accomplished when the angle ϕ n (t) reaches 2π or -2π, but the distance of the free end point to the origin is not fixed, leading to a one dimensional free parameter space. This undefined position is in favor of a winding time that is not too large when compared to any narrow escape problem where a Brownian particle has to find a small target in a confined domain [WaK93, HoS04, SSHE06, SSH07, BKH07, PWPSK09].

In this study, we consider not only that the initial condition satisfies |ϕ n (0)| < 2π, but we impose that ϕ n (0) is located far enough from 2π, to avoid studying any boundary layer effect, which would lead to a different MRT law. Indeed, starting inside the boundary layer for a narrow escape type problem leads to specific escape laws [START_REF] Singer | Narrow Escape I[END_REF]. Given a small ε > 0 we shall consider the space of configurations Ω ε such that |ϕ n (0)| < 2π -ε. We shall mainly focus on the stretched polymer, (θ 1 (0), θ 2 (0), . . . , θ n (0)) = (0, 0, . . . , 0) ,

and thus c n = n > 0 (in this case, ϕ n (0) = 0). Finally, it is quite obvious that winding occurs only when the condition:

nl 0 > L (13)
is satisfied, which we assume all along. The outline of our method is: first we show that the sum X n (t) converges (eq. ( 2)) and we obtain a Central Limit Theorem. Using Itô calculus, we study the sequence:

1 √ n X n (t) = 1 √ n [X n (t) -E (X n (t))] . (14)
and prove that 1 √ n X n (t), for n large, converges to a stochastic process which is a generalization of an Ornstein-Uhlenbeck process (GOUP), containing a time dependent deterministic drift c n e -t . This GOUP is driven by a martingale M (n) t , t ≥ 0 that we further characterize. Interestingly, the two cartesian coordinates of M (n) t , t ≥ 0 converge to two independent Brownian motions with two different time scale functions. To obtain an asymptotic formula for the MRT, we show that in the long time asymptotics, where winding occurs, the GOUP can be approximated by a standard Ornstein-Uhlenbeck process (OUP). Using some properties of the GOUP [START_REF] Vakeroudis | On hitting times of the winding processes of planar Brownian motion and of Ornstein-Uhlenbeck processes, via Bougerol's identity[END_REF], we finally derive the MRT for the polymer which is the mean time that |ϕ n (t)| = 2π.

3 Properties of the free polymer end X n (t) using a Central Limit Theorem

In this section we study some properties of the free polymer end X n (t). In particular, using a Central Limit Theorem, we show that the limit process satisfies a stochastic equation of a new type. To study the random part of X n (t), we shall remove from it its first moment and we shall now consider the asymptotic behavior of the drift-less sequence:

1 √ n X n (t) = 1 √ n [X n (t) -E (X n (t))] . (15)
We start by computing the first moment E (X n (t)). Because (see (2)), (θ i (t), t ≥ 0) are assumed to be n independent identically distributed (iid) Brownian motions with variance 2D, after rescaling, we obtain:

E (X n (t)) = E l + n k=1 e iB k (t) = l + n k=1 E e i(B k (t)-B k (0)) E e i(B k (0)) = l + c n e -t 2 . ( 16
)
where c n is defined by (7) and we have used that:

E e i(B k (t)-B k (0)) = e -t 2 . ( 17
)
We study the sequence (15) as follows:

1 √ n X n (t) = 1 √ n n k=1 e iB k (t) -E n k=1 e iB k (t) = 1 √ n n k=1 F k (t), (18)
where t) . Applying Itô's formula to

F k (t) = e iB k (t) -E e iB k (
Z (n) t = 1 √ n n k=1 F k (t) , (19) with Z
(n) 0 = 0, we obtain:

Z (n) t = i √ n t 0 n k=1 e iB k (s) dB k (s) - 1 2 √ n t 0 n k=1 e iB k (s) -E e iB k (s) ds = M (n) t - 1 2 t 0 Z (n) s ds, (20)
where:

M (n) t = i √ n t 0 n k=1 e iB k (s) dB k (s) = 1 √ n t 0 n k=1 (i cos(B k (s)) -sin(B k (s))) dB k (s) = -S (n) t + iC (n) t . (21)
We shall now study the asymptotic limit of the martingales

M (n) t
as n → ∞ and summarize our result in the following theorem; the convergence in law being considered there is associated with the topology of the uniform convergence on compact sets of the functions in C(R + , R 2 ).

Theorem 3.1 The sequence (M (n) t , t ≥ 0) converges in law to a Brownian motion in dimension 2, with 2 deterministic time changes. More precisely,

(S (n) t , C (n) t , t ≥ 0) (law) -→ n→∞ (σ ( 1 2 t 0 ds (1-e -2s )) , γ ( 1 2 t 0 ds (1+e -2s )) , t ≥ 0), (22)
where (σ u , γ u , u ≥ 0) are two independent Brownian motions.

Remark 3.2 a) The convergence in law ( 22) is a new result and it shows that the sum of the complex exponentials of i.i.d. Brownian motions can be approximated by a two dimensional Brownian motion, with a different time scale for each coordinate. b) An extension of Theorem 3.1 when (exp (iB(s)) , s ≥ 0) a Brownian motion on the unit circle is replaced by a BM on the unit sphere in R n is obtained in [START_REF] Holcman | A Central Limit Theorem for a sequence of Brownian motions in the unit sphere in R n[END_REF].

Proof of Theorem 3.1 See Appendix A.

We conclude that the sequence (M

(n) t , t ≥ 0) converges in law: (M (n) t , t ≥ 0) (law) -→ n→∞ (σ t 2 -1-e -2t 4
+ iγ t 2 + 1-e -2t 4 , t ≥ 0). (23)
The process (Z

(n)
t , t ≥ 0), which we defined in (20), is a generalization of the classical Ornstein-Uhlenbeck process; it is driven by (M (n) t , t ≥ 0) and, from (20), we obtain:

Z (n) t = e -t 2 t 0 e s 2 dM (n) s . ( 24
)
Corollary 3.3 a) The sequence (Z

(n) t , t ≥ 0) converges in law: Z (n) t (law) -→ n→∞ Z (∞) t , (25)
with:

Z (∞) t = e -t 2 t 0 sinh(s) dδ s + i e -t 2 t 0 cosh(s) d δs , (26)
where δ t , δt , t ≥ 0 are two independent 1-dimensional Brownian motions.

b)

Z (∞) t (law) -→ t→∞ 1 √ 2 N + i Ñ ,
where N and Ñ are two centered and reduced Gaussian variables.

Proof of Corollary 3.3 a) From ( 21), ( 22) and (24) we deduce: 28), we change variables: u = t -s and we obtain:

Z (n) t = e -t 2 t 0 e s 2 dM (n) s (27) (law) -→ n→∞ e -t 2 t 0 e s 2 1 -e -2s 2 √ sinh(s) dδ s + ie -t 2 t 0 e s 2 1 + e -2s 2 √ cosh(s) d δs ≡ Z (∞) t . (28) b) From (
Z (∞) t = t 0 e -t-s 2 1 -e -2s 2 dδ s + i t 0 e -t-s 2 1 + e -2s 2 d δs u=t-s = (law) t 0 e -u 2 1 2 - e -2(t-u) 2 dδ u + i t 0 e -u 2 1 2 + e -2(t-u) 2 d δu (29) L 2 -→ t→∞ 1 √ 2 ∞ 0 e -u 2 dδ u + i 1 √ 2 ∞ 0 e -u 2 d δu , (30)
where the two variables on the RHS of (30) are centered Gaussian with variance 1/2 and the convergence in L 2 for t → ∞ may be proved by using the dominated convergence theorem.

Remark 3.4 In [START_REF] Itô | Distribution-Valued Processes arising from Independent Brownian Motions[END_REF] and [START_REF] Ochi | Limit theorems for a class of Diffusion Processes[END_REF] there is a more complete asymptotic study for Z

(n) t (ϕ), as n → ∞, for a large class of functions ϕ, but we do not pursue this line of research here.

On one hand, from the identities (15), ( 16), ( 18) and ( 19), we obtain the following expansion:

X n (t) = X n (t) + E [X n (t)] = √ n Z (n) t + c n e -t 2 + l ⇒ Xn (t) ≡ X n (t) -l = √ n Z (n) t + c n e -t 2 , (31)
with n the number of rods/beads, Z

(n) t a GOUP driven by M (n) t
which is given by ( 23), c n = E n k=1 e iθ k (0) a constant depending on the mean initial configuration and l = L l 0 the rescaled distance of the fixed end from the origin 0 (l 0 is the fixed length of the rods). On the other hand, with Corollary 3.3(a), we deduce that:

Z (n) t ≡ 1 √ n X n (t) ≡ 1 √ n [X n (t) -E (X n (t))] = 1 √ n X n (t) -c n e -t 2 -l (law) -→ n→∞ e -t 2 t 0 sinh(s) dδ s + i e -t 2 t 0 cosh(s) d δs ≡ Z (∞) t . ( 32
)
4 Asymptotic expression for the MRT

We now study more precisely the different time scales of the two Brownian motions in ( 23) and (32). We estimate Z (∞) t for t large, the regime for which the rotation will be accomplished.

We introduce here the following notation:

L 2 ≈ denotes closeness in the L 2 -norm: for two stochastic processes (W (1) t , t ≥ 0) and (W (2) t , t ≥ 0), the notation W (1) t L 2 ≈ W (2) t means that lim t→∞ E W (1) t -W (2) t 2 = 0.
We shall show that, with B t = δ t + i δt , t ≥ 0 a 2-dimensional Brownian motion starting from 1:

Z (∞) t = e -t 2 t 0 sinh(s) dδ s + i e -t 2 t 0 cosh(s) d δs L 2 ≈ e -t 2 t 0 e s 2 √ 2 dB s . (33)
For this, it suffices to use the expression (30) and the following Proposition, which reinforces the

L 2
≈ result in (33). Thus, by multiplying both processes by e -t 2 , we obtain (33).

Proof of Proposition 4.1 The Gaussian martingale

t 0 sinh(s) -e s/2 √ 2 dδ s has increasing process t 0 sinh(s) - e s/2 √ 2 2 ds = t 0 e s 2 √ 1 -e -2s -1 2 ds ,
which converges as t → ∞. Hence, the limit variable

∞ 0 sinh(s) -e s/2 √ 2 dδ s
is Gaussian, and its variance is given by (we change variables: u = e -2s and B(a, b) denotes the Beta function with arguments a and b * * ):

∞ 0 ds e s 2 √ 1 -e -2s -1 2 = 1 4 1 0 du u -3/2 √ 1 -u -1 2 = 1 4 1 0 du u -3/2 (1 -u) -2 √ 1 -u + 1 = 1 4 B - 1 2 , 2 -2B - 1 2 , 3 2 -2 = π -3 2 .
To be rigorous, the integral

1 0 du u -α √ 1 -u -1 2
, which is well defined for 0 < α < 1, can be extended analytically for any complex α with Re(α) < 3.

For the convergence of the second process, it suffices to replace sinh(s) by cosh(s) ≡ e s 2 (1 + e -2s ). The limit variable

∞ 0 cosh(s) -e s/2
√ 2 d δs is also Gaussian, and, repeating the previous calculation, we easily compute its variance.

Asymptotic expression for the MRT

For t large, we derive an asymptotic value for the MRT. First, from (24) and (31):

Xn (t) = √ nZ (n) t + c n e -t 2 = √ ne -t 2 cn + t 0 e s 2 dM (n) s , (34)
where, the sequence cn is:

cn ≡ c n √ n , (35)
n is the number of rods/beads, and c n ≡ E n k=1 e iθ k (0) is a constant depending on the mean initial configuration. Thus, from (31), (33), (34) and also using the scaling property of Brownian motion: where

Xn (t) (law) ≈ n:large √ nY (n) t , (
Y (n) t ≡ e -t 2 cn + t 0 e s 2 dB s/2 . ( 37
)

Changing time and expression of the MRT

To express the MRT, we now apply deterministic time changes. To make our writing simple, we denote Y t for Y

(n)

t , and changing variables u = s 2 in (37), we obtain:

Y 2t = e -t cn + t 0 e u dB u . (38)
Now, there is another BM ( Bt , t ≥ 0), starting from cn , such that:

Y 2t = e -t Bαt , (39)
where:

α t = t 0 e 2s ds = e 2t -1 2 , hence: α -1 (t) = 1 2 ln (1 + 2t) . (40)
Applying Itô's formula to (39), we obtain:

dY 2s = -e -s
Bαs ds + e -s d Bαs .

We divide by Y 2s and we obtain:

dY 2s Y 2s = -ds + d Bαs Bαs .
Thus:

Im dY 2s Y 2s = Im d Bαs Bαs ,
which means that, if we denote:

θ Z t ≡ Im( t 0 dZ s Z s ), t ≥ 0, (41)
the continuous winding process associated to a generic stochastic process Z, then:

θ Y 2t = θ B αt .
Thus, the first hitting times of the symmetric conic boundary of angle c

T |θ Y | c ≡ inf t ≥ 0 : θ Y t = c , (42)
and

T |θ B| c ≡ inf t ≥ 0 : θ B t = c , (43)
for an Ornstein-Uhlenbeck process Y with parameter λ = 1, and for a Brownian motion B respectively, with relation (40), satisfy:

2T |θ Y | c = 1 2 ln 1 + 2T |θ B| c . (44)
Finally,

E 2T |θ Y | c = 1 2 E ln 1 + 2T |θ B| c (45) = ln 2 2 + 1 2 E ln T |θ B| c + 1 2 , (46)
and equivalently:

E T |θ Y | c = ln 2 4 + 1 4 E ln T |θ B| c + 1 2 . ( 47
)
Thus, by taking c = 2π, for n large, with φn (t) denoting the total angle of Xn (t), the mean time E [τ n ], where τn ≡ inf{t > 0, | φn (t)| = 2π}, that Xn (t) rotates around 0, is:

E [τ n ] ≈ √ n 4 ln 2 + E ln T |θ B| 2π + 1 2 . ( 48
)
By using the series expansion of log(1 + x), we obtain informally:

E ln T |θ B| c + 1 2 -E ln T |θ B| c = E ln 1 + 1 2T |θ B| c (49) = 1 2 E 1 T |θ B| c - 1 8 E   1 T |θ B| c 2   + . . . , (50)
where this expansion should be understood as follows: we obtain an alternate series of negative moments of T |θ B| c , which converges [START_REF] Vakeroudis | Integrability properties and Limit Theorems for the first exit times from a cone for planar Brownian motion[END_REF]. For our purpose, we will truncate the series. We shall estimate and use only the first moment of 1/T |θ B| c . By numerical calculations, the other moments can be neglected.

From the skew product representation [START_REF] Karatzas | Brownian Motion and Stochastic Calculus[END_REF][START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF] there is another planar Brownian motion (β u + iγ u , u ≥ 0), starting from log cn + i0, such that:

log Bt + iθ t ≡ t 0 d Bs Bs = (β u + iγ u ) u=Ht≡ t 0 ds | Bs| 2 , (51)
and equivalently:

log Bt = β Ht ; θ t = γ Ht . (52)
Thus, with

T |γ| c ≡ inf {t ≥ 0 : |γ t | = c} and T |θ B| c ≡ inf t ≥ 0 : θ B t = c , because θ T |θ B| c = γ H T |θ B| c : T |γ| c = H T |θ B| c , hence T |θ B| c = H -1 u u=T |γ| c
, where:

H -1 u ≡ inf{t : H t > u} = u 0 ds exp(2β s ) ≡ A u , (53)
and for u = T |γ| c , we obtain:

T |θ B| c = A T |γ| c . (54)
It is of interest for our purpose to look at the first negative moment of A t which has the following integral representation for any t > 0 ([Duf00], [D-MMY00], p.49, Prop. 7, formula (15)):

E 1 A t = ∞ 0 ye -y 2 t/2 coth π 2 y dy. (55)
With

β s = log(c n ) + β (0)
s , where (β

(0)
s , s ≥ 0) is a one-dimensional Brownian motion starting from 0, we obtain:

T |θ B| c = H -1 (T |γ| c ) ≡ T |γ| c 0 ds exp (2β s ) = (c n ) 2 T |γ| c 0 ds exp 2β (0) s ≡ (c n ) 2 T |θ B (1) | c , (56)
where

T |θ B (1) | c ≡ inf t ≥ 0 : θ B (1)
we get:

E 1 T |θ B (1) | c = ∞ 0 yE e -y 2 2 T |γ| c coth π 2 y dy = ∞ 0 y cosh(yc) coth π 2 y dy ≡ G(c). (58)
For c = 2π, we obtain the numerical result:

G(2π) ≡ E 1 T |θ B (1) | 2π ≈ 0.167. ( 59
)
Thus, with (56), and from (50), we have:

E ln T |θ B| c = 2 ln(c n ) + E ln T |θ B (1) | c , (60)
E ln T |θ B| 2π + 1 2 ≈ 2 ln(c n ) + E ln T |θ B (1) | 2π + 1 2c 2 n E 1 T |θ B (1) | 2π . (61)
For the first moment of ln T

|θ B| c
, for an angle c, using Bougerol's identity [START_REF] Ph | Exemples de théorèmes locaux sur les groupes résolubles[END_REF][START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF], there is the integral representation [START_REF] Comtet | Exponential functionals of Brownian motion and disordered systems[END_REF][START_REF] Vakeroudis | On hitting times of the winding processes of planar Brownian motion and of Ornstein-Uhlenbeck processes, via Bougerol's identity[END_REF]:

E ln T |θ B (1) | c = 2F (c) + ln (2) + c E , (62)
where † † :

F (c) = ∞ 0 dz cosh πz 2 ln (sinh (cz)) , (63)
and c E ≈ 0.577 denotes Euler's constant. For c = 2π, we have F (2π) ≈ 3.84. In Figure 2 we plot F with respect to the angle c. † † We note that there is a simple relation between F and G:

π 2 4c F ′ π 2 4c = c G(c).
For a more complete discussion, see [START_REF] Vakeroudis | On hitting times of the winding processes of planar Brownian motion and of Ornstein-Uhlenbeck processes, via Bougerol's identity[END_REF].

In summary, from (47), ( 59), ( 61) and (62), we approximate E T

|θ Y | 2π : E T |θ Y | 2π ≈ 1 4 2 ln(c n ) + Q + 1 2c 2 n E 1 T |θ B (1) | 2π , (64)
where:

Q = 2F (2π) + 2 ln 2 + c E ( 65
)
is a constant with F (2π) ≈ 3.84, c E ≈ 0.577, and

E 1 T |θ B (1) | 2π
≈ 0.167, thus:

Q ≈ 9.54. ( 66
)
Thus, by taking c = 2π, for n large, the mean time E [τ n ] that Xn (t) rotates around 0, is:

E [τ n ] ≈ √ n 4 2 ln(c n ) + 0.08 c2 n + Q , ( 67
)
and since:

cn ≡ c n √ n , (68)
the MRT of Xn (t) is given by the formula:

E [τ n ] ≈ √ n 4 2 ln c n √ n + 0.08 n c 2 n + Q . ( 69
)
For a long enough polymer, such that nl 0 >> L ⇒ n >> l, thus l = L l 0 is negligible with respect to X n (t) and from (31) X n (t) ≈ Xn (t). For a mean initial configuration c n ≡ E n k=1 e iθ k (0) , we obtain that the MRT of the polymer is given by the formula:

E [τ n ] ≈ √ n 4 2 ln E n k=1 e iθ k (0) √ n + 0.08n (E ( n k=1 e iθ k (0) )) 2 + Q . (70)
Finally, using the unscaled variables (4), we obtain: Expressions (70) and (71) show that the leading order term of the MRT depends on the initial configuration, however this dependence is weak.

E [τ n ] ≈ √ n 8D    2 ln   E n k=1 e i √ 2D θ k (0) √ n   + 0.08n E n k=1 e i √ 2D θ k (0) 2 + Q    . (71
We consider now that the polymer is initially stretched (θ k (0) = 0, ∀k = (1, ..., n)), hence c n = n. Thus, from (71), the MRT is approximately:

E [τ n ] ≈ √ n 8D ln (n) + 0.08 1 n + Q . (72)
In order to check the range of validity of formula (72), we ran some Brownian simulations. In Figure 3, we simulated the MRT with a time step dt = 0.01 for n = 50 to 300 rods in steps of 5, and for each n, we took 300 samples and averaged over all of them. The parameters we chose were D = 10 for the diffusion coefficient, L = 0.3 for the distance from the origin 0 and l 0 = 0.25 for the length of each rod, and for the initial condition we chose a stretched polymer located on the half line -→ 0x, θ k (0) = 0, ∀k = (1, ..., n) (hence c n = L + nl 0 ), and then we computed the MRT E [τ n ]. In Figure 4, we plot both the results from Brownian simulations and the formula (72). We considered values from n = 50 to 300 rods in steps of 10, D = 10, L = 0.3, l 0 = 0.25, thus the condition √ n l 0 >> L is satisfied. For each numerical computation, we performed 300 runs with a time step dt = 0.01. By comparing the numerical simulations (Fig. 4), with the analytical formula for the MRT, we see an overshoot. The straight initial configuration is no restriction to the generality of our study: we have c n ≤ n and the upper bound is achieved for a straight initial configuration. In Figure 5, we simulate the MRT E [τ n ] for both a random and an initially straight configuration (Brownian simulations with the same values for the parameters as above, after formula (72)).

Uniformly distributed initial angles

When the initial angles (θ 1 (0), θ 2 (0), . . . , θ n (0)) are uniformly distributed over [0, 2π], by averaging over all possible initial configurations, from (7) and (8), we obtain:

c n = E n k=1 e iθ k (0) = E n k=1 (cos(θ k (0)) + i sin(θ k (0))) = 0, (73)
hence, from ( 16),

E [X n (t)] ≡ l + c n = l. (74)
We define: We know that, for n large, with φn (t) denoting the total angle of Xn (t) the mean time E [τ n ], where τn ≡ inf{t > 0, | φn (t)| = 2π}, that Xn (t) rotates around 0, is:

Xn (t) ≡ 1 √ n X n (t) -l + 1 = 1 + Z (n) t . (75
E [τ n ] ≈ √ n 8D Q.
Finally, for a long enough polymer, such that nl 0 >> L ⇒ n >> l and n k=1 e iθ k (t) >> √ n, thus l = L l 0 and √ n are negligible with respect to X n (t)

and from (75) X n (t) ≈ Xn (t). Using the unscaled variables (4), with Q ≈ 9.62, the MRT satisfies:

E [τ n ] ≈ E [τ n ] ≈ √ n 8D Q. ( 76
)
Remark 4.2 Formula (71) or formulas (72) and (76) provide the asymptotic expansion for the MRT when θ 1 ∈ R + . In fact, θ 1 is a reflected Brownian motion in [0, 2π], thus a better characterization is to estimate the MRT by using the probability density function for θ 1 in the one dimensional torus [0, 2π]. In the Appendix of [START_REF] Vakeroudis | Nombres de tours de certains processus stochastiques plans et applications à la rotation d'un polymère[END_REF], we derive this probability density function and by repeating the previous calculations, we show that, e.g. formula (76), remains valid.

The Minimum Mean First Rotation Time

The Minimum Mean Rotation Time (MMRT) is the first time that any of the segments of the polymer loops around the origin,

MMRT ≡ min En E{τ n |c n } ≡ E [τ min ] , (77)
where E n is the ensemble of rods which can travel up to the origin. The MMRT is now a decreasing function of n. In Figure 6, we present some simulations for the MMRT as a function of n (100 simulations per time step dt = 0.01 with n = 4 to 15 rods, D = 10 for the diffusion constant , L = 0.3 for the distance from the origin 0 and l 0 = 0.25 the length of each rod). The initial configuration is such that |ϕ n (0)| < 2π -ε, e.g. the straight initial configuration: θ k (0) = 0, ∀k = (1, ..., n). In Figure 7, we present some Brownian simulations for the MMRT as a function of D and of L (

Initial configuration in the boundary layer

When the polymer has initially almost made a loop, we expect the MRT to have a different behavior. We look at this numerically, and we start with an initial polymer configuration in the boundary layer :

2π -ε ≤ |ϕ n (0)| < 2π.
where ϕ n is defined in eq. (3). The rotation of the polymer will be completed very fast and using Brownian simulations (100 runs per point with n = 10 rods and D = 1 for the diffusion constant, L = 0.1 for the distance from the origin 0 and l 0 = 0.2 for the length of each rod), we plotted in Figure 8 the results showing that when the initial total angle ϕ n (0) tends to 2π, the MRT tends to zero, with the precise asymptotic remaining to be completed. Numerically, we postulate that there is a threshold for an initial total angle the origin, while the other end moves as a Brownian motion. Interestingly, we have shown here that the motion of the free polymer end satisfies a new stochastic equation (31), containing a nonlinear time-dependent deterministic drift. When n is large, the limit process is an Ornstein-Uhlenbeck process, with different time scales for each of the two coordinates.

|ϕ n (0)| = π 2 . When |ϕ n (0)| < π 0 pi/4 pi/2 3pi/4 pi 5pi/4 3pi/2 7pi/4 2pi
We found that the MRT E [τ n ] actually depends on the mean initial configuration of the polymer. This result is in contrast with the one of the small hole theory [WaK93, WHK93, HoS04, SSHE06, SSH06a, SSH06b, SSH07, SSH08] where the leading order term of the MFPT for a Brownian particle to reach a small hole does not depend on the initial configuration. Although the MRT is not falling exactly into the narrow escape problems, it is a rare event and the polymer completes a rotation when the free end reaches any point of the positive x-axis. The reason why the initial configuration survives in the large time asymptotic regime, is due to the dynamics of the free moving end, approximated as a sum of i.i.d. variables, which is not Markovian. It leads to a process with memory. In summary, for nl 0 >> L, the leading order term of the MRT is given by: 1. for a general initial configuration:

E [τ n ] ≈ √ n 8D 2 ln c n √ n + 0.08 n c 2 n + Q , where c n ≡ E n k=1 e i √ 2D
θ k (0) , (θ k (0), 1 ≤ k ≤ n) is the sequence of the initial angles and Q ≈ 9.54, 2. for a stretched initial configuration:

E [τ n ] ≈ √ n 8D (ln(n) + Q) ,
3. for an average over uniformly distributed initial angles:

E [τ n ] ≈ √ n 8D Q,
where Q ≈ 9.62.

As we have shown, these formulas are in very good agreement with Brownian simulations (Fig. 4). After completion of this work, several questions arise naturally, namely:

-When the polymer has already made one loop, what is the probability that it makes a second loop before unwrapping and in that case, what is the MRT? -How can we extend our study in dimension 3 and higher dimensions? See [START_REF] Holcman | A Central Limit Theorem for a sequence of Brownian motions in the unit sphere in R n[END_REF] for some first results.

A Appendix: Proof of Theorem 3.1

We show the convergence of the sequence (M (n) t , t ≥ 0) to a Brownian motion in dimension 2, with a different time scale function for each coordinate. This classically involves 2 steps (see e.g. [START_REF] Billingsley | Convergence of Probability Measures[END_REF][START_REF] Billingsley | Probability and Measure[END_REF] or [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF] (Chapter XIII)):

1. the convergence of the finite dimensional distributions, and 2. the tightness of the sequence (M Consequently, with < M > t denoting the quadratic variation [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF] of the martingale (M t , t ≥ 0), we obtain:

S (n) t =
where the classical ergodic theorem yields the a.s. convergence [START_REF] Billingsley | Convergence of Probability Measures[END_REF]. We now give explicit expressions for the 3 right-hand sides.

Consider:

E (exp(iB 1 (s))) 2 = E [exp(2iB 1 (s))] = e -2s ,
from which we deduce:

E cos 2 (B 1 (s)) -sin 2 (B 1 (s)) = E [cos(2B 1 (s))] = e -2s
, and E [sin(2B 1 (s))] = 0.

Finally, we obtain:

E cos 2 (B 1 (s)) = 1 + e -2s 2 , (81)
E sin 2 (B 1 (s)) = 1 -e -2s 2 . ( 82
)
b) The previous results allow to obtain the convergence in law for M

(n) t 1 , . . . , M

(n) t k , say, as n → ∞. Indeed, by taking f, g : R + → R, simple functions, we may write:

E e i ∞ 0 f (u)dS (n) u + ∞ 0 g(u)dC (n) u ≡ E exp i Σ (n) ∞ , (83)
where:

Σ (n) t = t 0 f (u)dS (n) u + g(u)dC (n) u , (84)
as follows:

E exp i Σ (n) ∞ = E exp i Σ (n) ∞ + 1 2 < Σ (n) > ∞ exp - 1 2 < Σ (n) > ∞ .( 85
)
The results obtained in part a), together with the fact that [ReY99]:

E exp i Σ (n) ∞ + 1 2 < Σ (n) > ∞ = 1 (86)
now yield:

E exp i Σ (n) ∞ -→ n→∞ exp - 1 2 ∞ 0 f 2 (u)
1 -e -2u 2 + g 2 (u) 1 + e -2u 2 du .(87)

c) It now remains to prove the tightness [START_REF] Billingsley | Probability and Measure[END_REF] of the distributions of the sequence M (n) , which follows from a classical application of Kolmogorov's criterion; indeed, for β > 0 and c β a positive constant,

E M (n) t -M (n) s 2β ≤ c β E S (n) t -S (n) s β + E C (n) t -C (n) s β ≤ 2c β |t -s| β . (88)
We refer the reader who may want more details about the arguments used to [START_REF] Revuz | Continuous Martingales and Brownian Motion[END_REF], Chapter XIII, where convergence in distribution on the canonical space C(R + , R) is discussed.

Figure 1 :

 1 Figure 1: Schematic representation of a planar polymer winding around the origin.(a) A random configuration, (b) MRT when the n-th bead reaches 2π.

Proposition 4. 1

 1 As t → ∞, the Gaussian martingales t 0 sinh(s) dδ s -, t ≥ 0 converge a.s. and in L 2 . The limit variables are Gaussian with variances π-3 2 and -1 + 2 √ 2 -2a s (1) ≈ 0, 033, where a s (x) ≡ arg sinh(x) ≡ log(x + √ 1 + x 2 ), x ∈ R, respectively.

  36) * * We recall that if (Γ(x), x ≥ 0) denotes the Gamma function, then B(a, b) = Γ(a)Γ(b) Γ(a+b) .

t=

  c is the first hitting time of the symmetric conic boundary of angle c of a Brownian motion B (1) starting from 1+i0. Hence, from (54-55) and by using the Laplace transform for the hitting time T |γ| c [ReY99] (Chapter II, Prop. 3.7) or [PiY03](p.298):

FFigure 2 :

 2 Figure 2: F as a function of the angle c.

Figure 3 :

 3 Figure 3: MRT of the free polymer end as a function of the number of beads n (Brownian simulations).

Figure 4 :

 4 Figure 4: Comparing the Analytical Formula and the Brownian simulations of the MRT. The MRT of the polymer end is depicted with respect to the number of beads n.

Figure 5 :

 5 Figure 5: MRT of the free polymer end as a function of the number of beads n for the straight and for a random initial configuration (Brownian simulations).

Figure 6 :

 6 Figure 6: Brownian simulations of the MMRT as a function of the number of beads n -2 for D = 10.

Figure 7 :

 7 Figure 7: Brownian simulations of the MMRT as a function of the number of beads n -2. (a) For several values of D with L = 0.3, (b) for several values of L with D = 10.

Figure 8 :

 8 Figure 8: Brownian simulations of the MRT as a function of the initial total angle ϕ n (0), where ϕ n is defined in (3).

  k (s)) dB k (s).

EEE

  sin 2 (B 1 (s)) ds, (78) cos 2 (B 1 (s)) ds, (79)S (n) , C (n) [sin(2B 1 (s))] ds,

, the MRT appears to be independent from this angle, whereas for |ϕ n (0)| > π 2 , the MRT decreases to zero. However, this needs further investigation.5 Discussion and conclusionIn the present paper, we studied the MRT for a planar random polymer consisting of n rods of length l 0 . The first end is fixed at a distance L from