Finite-time and finite-size scalings in the evaluation of large-deviation functions: Numerical approach in continuous time

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1 PHYSICAL REVIEW E 95, (2017) Finite-time and finite-ize caling in the evaluation of large-deviation function: Numerical approach in continuou time Eteban Guevara Hidalgo, 1,2,* Takahiro Nemoto, 2,3, and Vivien Lecomte 2,4, 1 Intitut Jacque Monod, CNRS UMR 7592, Univerité Pari Diderot, Sorbonne Pari Cité, F , Pari, France 2 Laboratoire de Probabilité et Modèle Aléatoire, Sorbonne Pari Cité, UMR 7599 CNRS, Univerité Pari Diderot, Pari, France 3 Philippe Meyer Intitute for Theoretical Phyic, Phyic Department, École Normale Supérieure and PSL Reearch Univerity, 24 rue Lhomond, Pari Cedex 05, France 4 LIPhy, Univerité Grenoble Alpe and CNRS, F Grenoble, France (Received 29 July 2016; revied manucript received 10 May 2017; publihed 28 June 2017) Rare trajectorie of tochatic ytem are important to undertand becaue of their potential impact. However, their propertie are by definition difficult to ample directly. Population dynamic provide a numerical tool allowing their tudy, by mean of imulating a large number of copie of the ytem, which are ubjected to election rule that favor the rare trajectorie of interet. Such algorithm are plagued by finite imulation time and finite population ize, effect that can render their ue delicate. In thi paper, we preent a numerical approach which ue the finite-time and finite-ize caling of etimator of the large deviation function aociated to the ditribution of rare trajectorie. The method we propoe allow one to extract the infinite-time and infinite-ize limit of thee etimator, which a hown on the contact proce provide a ignificant improvement of the large deviation function etimator compared to the tandard one. DOI: /PhyRevE I. INTRODUCTION Rare event and rare trajectorie can be analyzed through a variety of numerical approache, ranging from importance ampling [1] and adaptive multilevel plitting [2] to tranition path ampling [3] (ee, e.g., Ref. [4,5] for review). In thi paper, we focu on population dynamic algorithm, a introduced in Ref. [6,7], which aim at tudying rare trajectorie by exponentially biaing their probability. Thi make it poible to render typical the rare trajectorie of the original dynamic in the imulated dynamic. The idea i to perform the numerical imulation of a large number of copie of the original dynamic, upplemented with election rule which favor the rare trajectorie of interet. The verion of the population dynamic algorithm introduced by Giardinà, Kurchan, and Peliti [6] provide a method to evaluate the large deviation function (LDF) aociated to the ditribution of a trajectory-dependent obervable. The LDF i obtained a the exponential growth rate that the population would preent if it wa not kept contant [8]. Under thi approach, the correponding LDF etimator i in fact valid only in the limit of infinite imulation time t and infinite population ize. The uual trategy that i followed in order to obtain thoe limit i to increae the imulation time and the population ize until the average of the etimator over everal realization doe not depend on thoe two parameter, up to numerical uncertaintie. The limitation and aociated improvement of the population dynamic algorithm have been tudied in Ref. [9 12]. In thi paper, following a different approach, we propoe an original and imple method that take into account the exact caling of the finite-t and finite- correction in order to provide ignificantly better LDF etimator. * eteban_guevarah@hotmail.com nemoto@lpt.en.fr vivien.lecomte@univ-grenoble-alpe.fr In Ref. [13], we performed an analytical tudy of a dicretetime verion of the population dynamic algorithm. We derived the finite- and finite-t caling of the ytematic error of the LDF etimator, howing that thee behave a 1/ and 1/t in the large- and large-t aymptotic repectively. In principle, knowing the caling apriorimean that the aymptotic limit of the etimator in the t and limit may be interpolated from the data at finite t and. However, whether thi idea i actually ueful or not i a nontrivial quetion, a there i alway a poibility that onet value of and t caling are too large to ue thee caling. In the preent paper, we conider a continuou-time verion of the population dynamic algorithm [14,15]. We how numerically that one can indeed make ue of thee caling propertie in order to improve the etimation of LDF, in an application to a ytem with many-body interaction (a contact proce). We illutrate in Fig. 1 the improvement in the determination of the LDF etimator. We emphaize that the two verion of the algorithm differ on a crucial point, which make it impoible for an extenion of the analyi developed in Ref. [13] to be done traightforwardly in order to comprehend the continuou-time cae (ee Appendix A). We thu tre that the obervation of thee caling themelve i alo nontrivial. The paper i organized a follow. In Sec. II, we introduce the continuou-time cloning algorithm. We define the large deviation of the additive obervable of interet and we detail how to etimate them. In Sec. III A, we tudy the behavior of the LDF etimator a a function of the duration of the obervation time (for a fixed population ) and we ee how it infinite-time limit can be extracted for the numerical data. In Sec. III B, we analyze the behavior of the etimator a we increae the number of clone (for a given final imulation time) and the infinite-ize limit of the LDF etimator. Baed on thee reult, we preent in Sec. IV, a method which allow u to extract the infinite-time, infinite-ize limit of the large deviation function etimator from a finite-time, finite-ize caling analyi. Our concluion are made in Sec. V. In order /2017/95(6)/062134(10) American Phyical Society

2 GUEVARA HIDALGO, NEMOTO, AND LECOMTE PHYSICAL REVIEW E 95, (2017) Relative Sytematic Error Standard Etimator Improved Etimator FIG. 1. Relative ytematic error [ () ψ()]/ψ() between the numerical etimator () and the analytical LDF ψ(). The error for the tandard etimator (N (T ) i hown in filled circle and for the improved one, f [Eq. (23)] in empty circle. The caling method propoed in thi paper wa teted on the contact proce (ee Sec. IIE2)(withL = 6, h = 0.1, and λ = 1.75) for a et of population ={20,...,200}, a imulation time T = 100, and R = 1000 realization. A can be een, the error due to finite-ize and finite-time effect can be reduced through the improved etimator. to complement the main dicuion done through the paper we alo preent the following: In Appendix A, an analyi of the difficulty of an analytical approach to the continuou-time algorithm. Then, in Appendix B, an alternative way of defining the LDF etimator i dicued. Finally, in Appendix C, we tudy the fluctuation of the LDF etimator. II. CONTINUOUS-TIME CLONING ALGORITHM A. Large deviation of additive obervable We conider a general Markov proce on a dicrete pace of configuration {C}, with tranition rate W(C C ). The probability P (C,t) for the ytem to be in a configuration C at time t verifie a mater equation of the form t P = WP, where the mater operator W i a matrix of element (W ) C C = W(C C ) r(c)δ CC (1) and where r(c) = C W(C C ) i the ecape rate from configuration C. A trajectory of configuration generated in thi proce i denoted by (C 0,...,C K ), tarting from C 0 and preenting K jump occurring at time (t k ) 1 k K.We denote by C(t ) the tate of the ytem at time t : When t k t <t k+1, C(t ) = C k (k = 0,1,2,...,K 1) with t 0 = 0. We are epecially intereted in the large deviation of additive obervable of the form K 1 t O = a(c k,c k+1 ) + dt b[c(t )], (2) k=1 0 for trajectorie of fixed duration t. The function a and b decribe the elementary increment of the obervable: a account for quantitie aociated with tranition (of tate), wherea b doe for tatic quantitie. A imple example of obervable of thi form i that of the activity O = K, which i the number of configuration change on the time interval [0,t] (in thi cae one ha a(c,c ) = 1 and b 0). We denote the joint ditribution function of the tate C and thee obervable O at time t by P (C,O,t). In order to analyze large deviation of thee additive obervable, we follow the tandard procedure a explained, for example, in Ref. [14,15]. For thi, we conider the moment generating function Z(,t) = e O, (3) where i the expected value with repect to trajectorie of duration t. The parameter biae the tatitical weight of hitorie and fixe the average value of O, o that 0 favor it nontypical value. Since the obervable O i additive and the ytem i decribed by a Markov proce, Z(,t) atifie at large time the caling Z(,t) e tψ() for t, (4) where ψ() i the growth (or decay) rate of Z(,t) with repect to time. Thi exponent, known a the caled cumulant generating function (CGF), i the quantity of interet in thi paper. It allow one to recover the large-time limit of the cumulant of O a derivative of ψ() in = 0, and more generically, the ditribution of O/t from the Legendre tranform of ψ() [16], known a a (large deviation) rate function. Hereafter, we ue the term large deviation function to refer both to the CGF and to the rate function by auming thee two are equivalent. Note that thi equivalence i at leat atified in ytem that do not how any phae tranition (a ingularity in the rate function). B. The mutation-election mechanim The moment generating function Z(,t) can be computed numerically uing the cloning algorithm [6,7]. In order to do that, we introduce the Laplace tranform of the probability ditribution P (C,O,t), defined a P ˆ(C,,t) = do e O P (C,O,t). (5) Thi Laplace tranform allow u to recover the moment generating function a Z(,t) = ˆ C P (C,,t). The probability P ˆ(C,,t) atifie an -modified mater equation for it time evolution (ee, e.g., Ref. [17]), t P ˆ = W P, ˆ (6) where the -modified mater operator W i defined a (W ) C C = W (C C ) r (C)δ CC + δr (C)δ CC. (7) Here, δr (C) = r (C) r(c) b(c), W (C C ) = e a(c,c ) W(C C ), (8) and r (C) = W (C C ). C (9) Contrary to the original operator (1),the -modified operator (7) doe not conerve probability (ince δr (C) 0), implying that there i no obviou way to imulate (6). However, thi timeevolution equation can be interpreted not a the evolution of a ingle ytem but a a population dynamic on a large number of copie of the ytem which evolve in a coupled way [6,7]

3 FINITE-TIME AND FINITE-SIZE SCALINGS IN THE... PHYSICAL REVIEW E 95, (2017) More preciely, reading the operator of the modified mater equation (6)ain(7), we find that thi evolution equation can be een a a tochatic proce of tranition rate W (C C ) and a election mechanim of rate δr (C) = r (C) r(c) b(c), (10) where a copy of the ytem in configuration C i copied at rate δr (C) (ifδr (C) > 0) or killed at rate δr (C) (if δr (C) < 0). A detailed below, the CGF ψ() irecovered from the exponential growth (or decay) rate of a population evolving with thee rule. C. Continuou-time population dynamic (contant-population approach) The mutation-election mechanim we jut decribed can be performed in a number of way. One of them conit in keeping the total number of clone contant for each pre-fixed time interval (ee Ref. [6,13] for example). Another one, which we ue throughout thi paper, conit in performing thee election mechanim along with each evolution of the copie [5,14,15]. A detailed decription of thi approach i preented below. See alo Appendix A for a brief explanation about important difference between thee two technique. 1. The cloning algorithm We conider clone (or copie) of the ytem. The dynamic i continuou in time: For each copy, the actual change of configuration occur at time (which we call evolution time ) which are eparated by interval whoe duration i ditributed exponentially. At a given tep of the algorithm, we denote by t ={t (i) } i=1,...,nc the et of the future evolution time of all copie and by c ={c i } i=1,...,nc the configuration of the copie. Their initial configuration do not affect the reulting caled cumulant generating function in the large-time limit. However, for the concretene of the dicuion, without lo of generality, we aume that thee copie have the ame configuration C at t = 0. The cloning algorithm i contituted of the repetition of the following procedure. 1. Find the clone whoe next evolution time i the mallet among all the clone: Find j = argmin i t (i). 2. Compute y j = Y (c j ) + ɛ, where the cloning factor Y (c j ) i defined a e t(c j ) δr (c j ), t(c j ) i the time pent by the clone j in the configuration c j ince it lat configuration change, and ɛ i a random number uniformly ditributed on [0,1]. 3. If y j = 0, remove thi copy from the enemble, and if y j > 0, make y j 1 new copie of thi clone. 4. For each of thee y j copie (if any), the tate c j i changed independently to another tate c j, with probability W (c j c j )/r (c j ). 5. Chooe a waiting time t from an exponential law of parameter r (c j ) for each of thee copie. It next change of configuration will occur at the evolution time t (j) + t. 6. In order to keep the total number of copie contant, we chooe randomly and uniformly (i) a clone k, k j and we copy it (if y j = 0), or (ii) y j 1 clone and we erae them (if y j > 1). D. Cumulant generating function etimator The CGF etimator can be obtained from the algorithm we jut decribed from the exponential growth rate that the population would preent if it wa not kept contant [5]. More preciely, thi etimator i defined a = 1 K t log X i, (11) where X i = ( + y i 1)/ are the growth factor at each tep j of the procedure decribed above, and K i the total number of configuration change in the full population up to time t (which ha not to be confued with K). It i important to remark (a wa dicued in Ref. [8] in a noncontant population context) that thi growth rate can be alo computed from a linear fit over the recontructed log population and the initial tranient regime, where the dicretene effect are preent, can be dicarded in order to obtain a better etimation. In practice, in order to obtain a good etimation of the CGF, it i normal to launch the imulation everal time (where we denote by R the number of realization of the ame imulation) and to etimate the arithmetic mean of the obtained value of (11) over thee R imulation. Strictly peaking (a dicued in Sec. 3.2 of Ref. [8]), a the imulation doe not top exactly at the final imulation time T but at ome time tr F T (which i different for every r {1,...,R}), the average over R realization of i then correctly defined a = 1 R 1 K r log Xi r R. (12) t F r=1 r However, we have oberved that for not too hort imulation time, (T ) (tr F) i mall. By auming t r F T, Eq. (12) can be approximated by replacing tr F by T (which i what we do in practice). It i important to remark that the CGF etimator can be defined differently from Eq. (12). Thi i done by uing an alternative way of computing the average over R realization (for an example on thi topic, ee Appendix B). Equation (12) allow u to etimate the CGF uing the contant-population approach of the continuou-time cloning algorithm for a -biaed Markov proce, given a fixed number of clone, a imulation time T, and R realization of the algorithm. E. Example model In order to analyze the finite-time and finite- caling of the CGF etimator, we introduce two pecific model: a imple two-tate annihilation-creation dynamic and a contact proce on a one-dimenional periodic lattice [14,18]. In both cae, we conider the activity K a the additive obervable O and the analytical expreion of the CGF ψ() wa obtained by olving the larget eigenvalue of the operator W given by (7). Below we define thee model. 1. Annihilation-creation dynamic The dynamic occur in one ite where the only two poible configuration C are either 0 or 1. The tranition rate are i=1 i=1 W(0 1) = c, W(1 0) = 1 c, (13)

4 GUEVARA HIDALGO, NEMOTO, AND LECOMTE PHYSICAL REVIEW E 95, (2017) where c [0,1]. The analytical expreion for the CGF of the activity in thi cae correpond to ψ() = [1 4c(1 (1 e 2 )] 1/2. (14) 2. Contact proce Each poition i of a L-ite, one-dimenional lattice i occupied by a pin which i either in the tate n i = 0or n i = 1. The configuration C i then contituted by the tate of thee pin, i.e., C = (n i ) L i=1. The dynamic occur on thi lattice with periodic boundary condition with tranition rate W(n i = 1 n i = 0) = 1 and W(n i = 0 n i = 1) = λ(n i 1 + n i+1 ) + h, (15) where λ and h are poitive contant. Thi model i an example of contact procee [18], which have been tudied in many context epecially for the pread of infection [19]. It ha been known that the correponding CGF develop a ingularity in L, howing a dynamical phae tranition [14,20]. III. FINITE-TIME AND FINITE- BEHAVIOR OF CGF ESTIMATOR In thi ection, we focu on the annihilation-creation proce for a peculiar value of ( = 0.2), which i repreentative of the full range of on which we tudy large deviation. A. Finite-time caling Here, we tudy the large-time behavior of the CGF etimator, at fixed number of clone. Figure 2 preent the average over R = 10 4 realization of the CGF etimator a a function of the (imulation) time for given number of * * analytical FIG. 2. Average over R = 10 4 realization of the CGF etimator (N [Eq. (12)] a a function of duration t of the obervation window, for {10,100,1000} clone, for the annihilation-creation dynamic (13) with c = 0.3. The analytical expreion for the large deviation function ψ() [Eq. (14)] i hown with a black dahed line and the fitting function f (N t encoding the finite-t caling [Eq. (17)] are hown with continuou curve. The (a priori) bet etimation of the large deviation function (to which we refer a tandard etimator) i given by (N (t) at the larget imulation time T = 1000, which are hown with olid circle (at the right end of the figure). The extracted infinite-time limit f (N are hown a dotted line and quare ( = 10), diamond ( = 100), and circle ( = 1000). fit clone ={10,100,1000}. It i compared with the analytical value ψ()[eq.(14)], which i hown with a black dahed line. A can be een in Fig. 2 for a mall number of clone ( = 10), the CGF etimator i highly deviated from the analytical value ψ(). However, a and the imulation time t become larger, the CGF etimator get cloer to the analytical value ψ(). One can expect that in the t and limit, ψ() will be obtained from the etimator a lim lim t (N (t) = ψ(), (16) a it wa derived in Ref. [13]. However, in a practical implementation of the algorithm, thi infinite-time and infiniteize limit are not achievable and we ue large but finite imulation time t and number of clone. Thi fact motivate our analyi of the actual dependence of the etimator with t and. The tandard etimator of the large deviation function i the value of at the larget imulation time T and for the larget number of clone, (e.g., (T ) for = 1000 and T = 1000, the black olid circle in Fig. 2). Thi value provide the (a priori) bet etimation of the large deviation function that we can obtain from the continuou-time cloning algorithm. However encouragingly, a we detail later, thi etimation can be improved by taking into account the convergence peed of the CGF etimator. The reult of fitting (t) with the curve f (N t defined a f (N t f (N + b(n t t 1 (17) i hown with olid line in Fig. 2. The fitting parameter f (N and b (N t can be determined from the leat quare method by minimizing the deviation from (t). The clear coincidence between (t) and the fitting line indicate the exitence of a 1/t convergence of (t) to lim t (t) (that we call 1/t caling). Thi property can be derived from the aumption that the cloning algorithm itelf i decribed by a Markov proce: In Ref. [13] with a different verion of the algorithm, we contructed a meta-markov proce to decribe the cloning algorithm by expreing the number of clone by a birth-death proce. Once uch meta proce i contructed, the CGF etimator (11) i regarded a the time average of the obervable X i within uch meta-markov proce. 1 We now recall that time-averaged quantitie converge to their infinite-time limit with an error proportional to 1/t when the ditribution function of the variable converge exponentially (a in Markov procee). Thi lead to the 1/t caling of CGF etimator (17). We note that contructing uch meta-markov proce explicitly i not a trivial tak, and for the algorithm dicued here, uch a contruction remain a an open problem. By auming the validity of the caling form (17), it i poible to extract the infinite-time limit of the CGF etimator from finite-time imulation. We denote thi infinite-time limit 1 In other word, t (N i an additive obervable of the meta proce decribing the cloning algorithm, a read from (11)

5 FINITE-TIME AND FINITE-SIZE SCALINGS IN THE... PHYSICAL REVIEW E 95, (2017) analytical FIG. 3. CGF etimator (N (T ) [Eq. (12)] for given final (imulation) time T ={200,300,500,1000} a a function of the number of clone (on the range ). The analytical value [Eq. (19)] with continuou curve. A large imulation time for a mall number of clone, hown in (A), produce a better etimation compared to the one given by the larget number of clone with a relatively hort imulation time, which i hown in (B). The bet CGF etimation we can naively ψ()(14) i hown with a dahed line and the fit g (T ) obtain would be given by (N (T ) at larget imulation time T and larget number of clone. However, the extracted infinite-ize limit g (T ) provide a better etimation in comparion. Thee limit are hown with dotted line and circle (T = 200), croe (T = 300), diamond (T = 500), and dot (T = 1000). Additionally, c = 0.3 and = 0.2. a f (N and it i expected to be a the better etimator of CGF than (T ) at finite T, provided that f (N = lim t (N (t). (18) In Fig. 2, we how f (N with dotted line and circle ( = 10), diamond ( = 100), and quare ( = 1000). A can be een, thi parameter indeed provide a better numerical etimate of ψ() than (T ). B. Finite- caling Here, we tudy the behavior of the CGF etimator (T ) a we increae the number of clone, for a given final (imulation) time T. Similar to what we did in Sec. III A, we conider a curve in the form g (T ) = g (T ) + b (T ) Nc 1, (19) where g (T ) and b (T ) are fitting parameter which are determined by the leat quare fitting to (T ). The obtained g (T ) a a function of are hown in Fig. 3 a olid line. We conidered four value of final imulation time T ={200,300,500,1000} and population ize in the range A can be een, thee curve decribe well the dependence in of (T ), indicating that (T ) converge to it infinite- limit with an error proportional to 1/ (that we call 1/ caling). Thi caling could be proved under general aumption in Ref. [13], (i) however, without covering the continuou-time algorithm dicued here and fit (ii) for the CGF etimator (T ) conidered the T limit, intead of finite T. The generalization of the argument preented in Ref. [13] in order to cover the general cae (i) and (ii) i an important open direction of reearch. By auming the validity of uch 1/ caling, we can evaluate the limit of obtained from finite imulation a g (T ) (T ) a the fitting parameter g (T ) = lim (N (T ). (20) Thee parameter g (T ) (to which we refer a infinite-ize limit) are hown in Fig. 3 a dotted line. A hown in the figure, g (T ) provide better etimation of ψ() than the one given by the tandard etimator (T ). Complementary to the dicuion in thi ection, in Appendix C we analyze the fluctuation of the CGF etimator. IV. FINITE-TIME AND FINITE- SCALING METHOD TO ESTIMATE LARGE DEVIATION FUNCTIONS In the previou ection, we have hown how it i poible to extract f (N and g (T ) from finite T and finite imulation, repectively. In thi ection, we combine both of thee 1/tand 1/ -caling method in order to extract the infinite-time and infinite-ize limit of the CGF etimator. Thi limit give a better evaluation of the large deviation function within the cloning algorithm than the tandard etimator. We firt note that either of f (N or g (T ) i expected to converge to ψ() a or a T. We checked numerically thi property by defining the ditance D between ψ() and it numerical etimator, D [,ψ() ] = ( ) ψ(). (21) FIG. 4. Ditance D [Eq. (21)] between the analytical CGF ψ() and it numerical etimator (N, a a function of time t in log-log cale. The ditance are computed from the value in Fig. 2. Thi ditance behave a a power law of exponent 1 onatimewindow, where the ize of the time window increae a increae. Thi illutrate the caling (22). The parameter of the model are c = 0.3, =

6 GUEVARA HIDALGO, NEMOTO, AND LECOMTE PHYSICAL REVIEW E 95, (2017) a a function of a f (N f + b(n 1, (23) which mean that f (N itelf exhibit 1/ correction for large but finite. By uing thi caling, we detail below in Sec. IV A the method to extract the infinite-time infinite- limit of the CGF etimator (T ) from finite-time and finite- data. We note that thi method can be ued for a relatively hort imulation time and a relatively mall number of clone (ee Fig. 6). In Sec. IV B, we preent numerical example of the application of thi method to the contact proce. FIG. 5. Etimator of the large deviation function (N (t) aa function of time and the number of clone. The etimator (N (T ) at final imulation time T = 100 a a function of the number of clone (up to = 200) i hown a black circle. The bet CGF etimation under thi configuration given by the tandard etimator, i.e., (Nc=200) (T = 100) i hown a a yellow circle. The analytical value of the CGF ψ() i obtained from the larget eigenvalue of the operator (7) and hown a a black dahed line. The extracted limit f i hown with red quare. Additionally, L = 6, = 0.15, h = 0.1, λ = 1.75, and R = Thi quantity i hown in Fig. 4 a a function of t in log-log cale. A we can ee, a increae, log D behave a traight line with lope 1 on a time window which grow with.in other word, when, ψ() t 1. (22) Inpired by thi obervation, we aume the following caling for the fitting parameter f. If we conider a et of imulation performed at population ize ={ (1),..., (j) },the obtained infinite-time limit of the CGF etimator f behave (a) fit A. The caling method The procedure i ummarized a follow: 1. Determine the average over R realization (t) [Eq. (12)] up to a final imulation time T for each. 2. Determine the fitting parameter f (N defined in the form f (N t = f (N + b (N t t 1 from each of the obtained (t). 3. Determine f from a fit in ize f (N = f + b(n Nc 1 [Eq. (23)] on f (N. The reult obtained for f render a better etimation of ψ() than the tandard etimator (t) evaluated for = max and for t = T. B. Application to the contact proce We apply the caling method to the one-dimenional contact proce (ee Sec. II E for the definition). We et L = 6, h = 0.1, λ = 1.75, T = 100, and = A we detail below, we compare the improved etimator f obtained from the application of the caling method (for = {20,40,...,180,200}) with the tandard etimator (T ) (for = max = 200) analytical (b) fit FIG. 6. (a) Projection of the urface repreented in Fig. 5 over the plane t. (N (t) i repreented for = 20 and = 200 with blue dot. The etimation (N (T ) of the large deviation (at the final imulation time T = 100) are hown in large blue dot for all the value of conidered. The fit in time [Eq. (17)] over (N (t) i hown a black olid line (for = 20 and = 200) and dotted line (for other value of ). (b) Projection at the final imulation time T = 100 on the plane, (N (T ) i hown in large blue dot. The infinite-time limit f (N a a function of [ee Eq. (17)] i repreented in red circle. The reult of fitting (N (T ) [Eq. (19)] and f (N [Eq. (23)] are hown i hown with blue dahed line and diamond meanwhile the infinite-ize with blue and red olid curve repectively. The infinite- limit g (T ) and infinite-time limit f i hown with a red dotted line in both panel (a) and (b). The extracted limit f render a better etimate of the large deviation function than (Nc=200) (T = 100) (and alo than g (T ) ), demontrating the efficacy of the method propoed

7 FINITE-TIME AND FINITE-SIZE SCALINGS IN THE... PHYSICAL REVIEW E 95, (2017) Figure 5 repreent the behavior of the etimator (t)a a function of the imulation time t and of the number of clone. The value of the etimator at the final imulation time T are repreented with black circle for each and with a yellow circle for = max. The analytical expreion for the large deviation function ψ() i hown in a black dahed line. In Fig. 6(a), we how the projection of the urface of Fig. 5 on the plane t. The behavior in t of the etimator (t) i hown for = 20 and = 200, in blue dot in Fig. 6(a). The tandard CGF etimator, (T ), are hown in large blue dot in Fig. 6(a) (on the axi for T = 100). The fitting curve f (N t [Eq. (17)] are hown in black continuou line (for = 20 and = 200) and black dotted line (for other intermediate value of ). Next, we how in Fig. 6(b) the projection of the urface of Fig. 5 on the plane where the time ha been et to the larget t = T. The tandard CGF etimator, (T ) are plotted a blue filled circle, and the fitting curve g (T ) [Eq. (19)] on (T ) i hown a a blue olid line. From thee curve, we determine g (T ) (ee Sec. III B), which i hown a a blue dahed line and diamond. Finally, the parameter f (N extracted from the fitting on (t) (for each value of ) i hown a red circle in Fig. 6(b). Thee value alo cale a 1/ [Eq. (23)] and their fit i hown a a red olid curve. The caling parameter f obtained from thi lat tep provide a better etimation of the large deviation function than the tandard etimator (=200) (T = 100) that i widely ued in the application of cloning algorithm. Thi improvement i valid on a wide range of value of the parameter a can be viualized in Fig. 1, where we repreented the relative ytematic error [ () ψ()]/ψ() between the tandard and improved etimator () and the analytical LDF ψ(). V. CONCLUSION Direct ampling of the ditribution of rare trajectorie i a rather difficult numerical iue (ee, for intance, Ref. [21]) becaue of the carcity of the nontypical trajectorie. We have hown how to increae the efficiency of a commonly ued numerical method (the o-called cloning algorithm) in order to improve the evaluation of large deviation function which quantify the ditribution of uch rare trajectorie, in the large time limit. We ued the finite-ize and finite-time caling behavior of CGF etimator in order to propoe an improved verion of the continuou-time cloning algorithm, which provide more reliable reult, le affected by finitetime and finite-ize effect. We verified the reult oberved for the dicrete-time verion of the cloning algorithm [13] and we howed their validity alo for the continuou cae. Importantly, we howed how thee reult can be applied to more complex ytem. We note that the caling which rule the convergence to the infinite-ize and infinite-time limit (with correction in 1/ and in 1/t) have to be taken into account properly: Indeed, a power law, they preent no characteritic ize and time above which the correction would be negligible. The ituation i very imilar to the tudy of the critical depinning force in driven random manifold: The critical force preent correction in one over the ytem ize [22], which ha to be conidered properly in order to extract it actual value. Generically, uch caling alo provide a convergence criterion to the aymptotic regime of the algorithm: One ha to confirm that the CGF etimator doe preent correction (firt) in 1/t and (econd) in 1/ with repect to an aymptotic value in order to enure that uch value doe repreent a correct evaluation of the LDF. It would be intereting to extend our tudy of thee caling to ytem preenting dynamical phae tranition (in the form of a nonanalyticity of the CGF), where it i known that the finite-time and finite-ize caling of the CGF etimator can be very hard to overcome [14]. In particular, in thi context, it would be ueful to undertand how the dynamical phae tranition of the original ytem tranlate into anomalou feature of the ditribution of the CGF etimator in the cloning algorithm. Thee phae tranition are normally accompanied with an infinite ytem-ize limit (although there wa a report of dynamical phae tranition without taking a uch limit [23]). To overcome thee difficultie (caued by a large ytem ize and/or by the preence of a phae tranition), it may be ueful to ue the adaptive verion of the cloning algorithm [24], which ha been recently developed to tudy uch phae tranition, with the caling method preented in thi paper. ACKNOWLEDGMENTS E.G. thank Khahayar Pakdaman for hi upport and dicuion. Special thank go to the Ecuadorian Government and the Secretaría Nacional de Educación Superior, Ciencia, Tecnología e Innovación, SENESCYT. T.N. gratefully acknowledge the upport of Fondation Science Mathématique de Pari EOTP NEMOT15RPO. V.L. acknowledge upport by the PEPS LAABS Inphyniti CNRS project, by the ERC Starting Grant No MALIG and by the ANR-15-CE Grant LSD. APPENDIX A: ISSUES ON AN ANALYTICAL APPROACH In a previou analytical tudy [13], we conidered a dicrete-time verion of the population dynamic algorithm, where a cloning procedure i performed every mall time interval t. We have proved the convergence of the algorithm in the large-, large-t limit, and we alo derived that the ytematic error of the LDF etimator (i.e., the deviation of the etimator from the deired LDF) decayed proportionally to 1/ and 1/t. From a practical point of view, however, the formulation ued there had one problem. In order to prove the reult, we took the large frequency limit of cloning procedure or, in other word, we took the t 0 limit. A rough etimate of the error due to noninfiniteimal t prove to be O( t). For a fater algorithm, it i better to take thi value to be larger, and indeed empirically, we expect that thi error to be very mall (or rather diappearing in the large-t, limit). However, the detailed analytical etimation of thi error i till an open problem. In the main part of thi current paper, from a different point of view, we conider the continuou-time verion of the population dynamic algorithm [14,15]. Here, the cloning i performed at each change of tate of a copy. The time interval

8 GUEVARA HIDALGO, NEMOTO, AND LECOMTE PHYSICAL REVIEW E 95, (2017) t which eparate thoe change of tate are noninfiniteimal, which mean that the formulation we ued in Ref. [13] cannot be applied to undertand it convergence. Furthermore, becaue thee time interval are of noncontant duration and tochatically ditributed, the continuou-time algorithm i more difficult to handle analytically than the dicrete-time verion. Intead of puruing the analytical tudy within the continuou-time algorithm, we perform a numerical tudy, and we how that the 1/ and 1/t caling are alo oberved for the continuou-time algorithm. Although the proof of thee caling are beyond the cope of the current paper, thee numerical obervation upport a conjecture that uch caling in large t and in large limit are generally valid in cloning algorithm to calculate large deviation function. APPENDIX B: A DIFFERENT CGF ESTIMATOR Normally, CGF etimator i defined a an arithmetic mean over many realization, a een in (12). Here we how that another definition of the CGF etimator can be ued, which indeed provide better reult than the one from the tandard etimator (in ome parameter range). We define a new etimator a = 1 Kr T log Xi r, i=1 (B1) where we note that the average with repect to realization are taken inide the logarithm. A we dicued in Sec. IV C of [13], thi etimator provide a correct value of CGF ψ() in the infinite-time, infinite- limit. Thi i thank to the fact that the ditribution of concentrate around ψ() in thoe limit (the o-called elf-averaging property). At any finite population, one can rewrite uing the large-time LDF principle (C4) afollow: log et (N = 1 T = 1 T log d e T [I Nc ( )+ ], (B2) (B3) which prove that in the large-t limit, (N [ = min INc ( ) + ], (B4) to be compared to = argmin I Nc ( ). (B5) On one hand, the definition (B1) amount to etimate ψ from the exponential growth rate of the average of the final-t population of many mall (noninteracting) iland, where the cloning algorithm would be operated. On the other hand, the etimator (12) amount to etimate ψ from growth rate of a large iland gathering the full et of the R population. The later i thu expected to be a better etimator of ψ() than the former becaue it correpond to a large population, where finite-ize effect are le important. A a conequence, the etimator appear apriorito be a wore etimator than of ψ(). However, a hown in Sec. IV C of Ref. [13], at mall and finite, a upplementary bia introduced by taking (B1) FIG. 7. Comparion between two different etimator of the large deviation function, (N [Eq. (12)] hown in dot and (N [Eq. (B1)] in circle, for the annihilation-creation dynamic (13). The analytical value ψ() [Eq. (14)] i hown with a dahed line. Here we have alo compared two different value of parameter = 0.2 (blue upper curve) and = 1 (black). Additionally, = 100, c = 0.4, T = 500, and R = 500. The etimator (N numerical evaluation of the CGF at mall. provide a better in fact compenate for the finite- ytematic error preented by (12), for a imple two-tate model. Namely, the error i O(Nc 1 )for(12) while it i O( 2 Nc 1 ) for(b1). Thi fact i illutrated on Fig. 7, where we how that at mall = 0.2, provide a better etimation of ψ() than, while at larger ( = 1) the two etimator yield a comparable error. APPENDIX C: FLUCTUATIONS OF CGF ESTIMATOR 1. Central limit theorem From relation (12), one can infer that the diperion of the ditribution of depend on the imulation time t. Thi determine whether or not a large number of realization R i required in order to minimize the tatitical error. In fact, a een in Fig. 8, the diperion of i concentrated around it mean value, which approache the analytical value ψ()a the imulation time and the number of clone increae. We numerically confirm that thee ditribution are well approximated by a Gauian ditribution P ( (N ) Ae 1 C 2 ( (N B) 2, (C1) where the parameter B i equal to (T ) and the parameter A and 1/C 2 are repectively of the order of Nc 1/2 and. A mathematical argument to explain thi obtained Gauian ditribution i given a follow: At any given time (not necearily at T ), let u perform the following recaling: ˆ = (N σ (N, (C2) where σ 2 i the variance of the R realization of (N. Then, it produce a collape of the ditribution P ( ˆ ), for any t and any (Fig. 9). We remark then that the CGF etimator (12) i an additive obervable of the hitory of the

9 FINITE-TIME AND FINITE-SIZE SCALINGS IN THE... PHYSICAL REVIEW E 95, (2017) 200 (a) (b) ( FIG. 8. Ditribution P ( (N )ofthecgfetimator (N for (a) = 10, (b) = 100, and ( = 1000 and for imulation time t [10,1000]. Each realization (R = 10 4 for each imulation time) i hown with gray dot; meanwhile it repective Gauian fit [Eq. (C1)] i hown with a dotted or a continuou curve. The diperion of (N i wider for horter imulation time and mall. The mean value of the ditribution converge to the theoretical value a the imulation time and the number of clone increae. population, which follow a Markov dynamic. Hence, the recaled etimator ˆ follow a tandard normal ditribution in the large time limit, according to the central limit theorem (CLT): P ( ˆ (N ) 1 = e 1 2 ( ˆ (N ) 2. (C3) 2π We note that thi check of the CLT allow u to enure if the teady tate of the population dynamic ha been reached (note that in general the typical convergence time to the teady tate i larger than the invere of the pectral gap of the biaed evolution operator [8]). By conidering the caling (C2), we focu only on the mall fluctuation of around. But in general, the ditribution function i not Gauian, and in that cae we need to conider a large deviation principle a below. 2. Logarithmic ditribution of CGF etimator Since i itelf an additive obervable of the dynamic of the enemble of clone [13], the ditribution of the CGF Standard Normal Ditribution FIG. 9. The ditribution function of the recaled variable ˆ (N [Eq. (C2)]. Compatible with the central limit theorem, a collape of the ditribution function into a tandard normal ditribution for different number of clone i oberved. etimator atifie itelf a large deviation principle P ( (N ) e ti Nc ( (N ), (C4) where I Nc ( ) i the rate function. Thi rate function could be evaluated in principle from the empirical ditribution P ( )a ( I Nc ( ) ) 1 t log P ( (N ) (C5) for a large t. Here we try to etimate the rate function from thi equation. The numerical etimation of the right-hand ide of the lat expreion at final imulation time T i hown in Fig. 10(a), where we have defined ( Iˆ Nc ( ) ) 1 t log P ( (N ) 1 + t log P ( (N ) (C6) o that Iˆ Nc ( ) = 0. In the ame figure, we alo how (T ) a vertical dotted line which correpond to the minima of the logarithmic ditribution Iˆ Nc ( ). A can be een, thee minima are diplaced toward the analytical value ψ() (hown with a dahed line) a.the logarithmic ditribution Iˆ Nc alo become more concentrated a increae. Next, in order to tudy thi decreaing of the width, we how a recaled logarithmic ditribution function (1/ ) Iˆ Nc ( ) in Fig. 10(b). The minimum converge to the analytical value ψ() (black dahed line) a. In the infinite-time, infinite-ize limit of, it would be thu compatible with a logarithmic ditribution function given by I ( (N ) = lim 1 1 lim t t log P ( (N (t)), (C7) which i hown (recaled) with black dot in Fig. 10(b). By performing the hift ˇ = ( ) we can ee in the inet of Fig. 10(b) the uperpoition of quadratic deviation of the numerical etimator around the minimum of I Nc ˆ (epecially for = 100,1000). Thi indicate the decreaing of the fluctuation of CGF etimator proportional with both of T and (ee Ref. [13] for more detailed explanation)

10 GUEVARA HIDALGO, NEMOTO, AND LECOMTE PHYSICAL REVIEW E 95, (2017) (a) (b) FIG. 10. (a) Logarithmic ditribution Iˆ Nc ( (N ) [Eq. (C6)]. Numerical evaluation were made for three fixed population ize {10,100,1000} with a fixed imulation time T = The logarithmic ditribution preent a maller width a increae. The average over R realization of the CGF etimator (N (T ) correpond to the minimum of Iˆ Nc ( (N ) (dotted line) and converge to the analytical value ψ() (dahed line) a. (b) Recaled logarithmic ditribution 1 ˆ I Nc ( (N ) a a function of (N andaafunctionof ˇ (N = ( (N (N ) (inet) for a final imulation time T = The obtained logarithmic ditribution i well approximated by a quadratic form, although thee large deviation are in general not quadratic [13]. Thi mean that the direct obervation dicued here cannot capture the large deviation of the CGF etimator (ee alo Ref. [21] for more detailed tudy of the direct etimation of rate function). However, we note that, for practical uage of the algorithm, we only conider mall fluctuation decribed by the central limit theorem, although thee large fluctuation might play an important role in more complicated ytem, uch a the one preenting dynamical phae tranition. [1] H. Kahn and T. E. Harri, National Bureau of Standard Applied Mathematic Serie 12, 27 (1951). [2] F. Cérou and A. Guyader, Stoch. Anal. Appl. 25, 417 (2007). [3] P. G. Bolhui, D. Chandler, C. Dellago, and P. L. Geiler, Annu. Rev. Phy. Chem. 53, 291 (2002). [4] J. Bucklew, Introduction to Rare Event Simulation (Springer, Berlin, 2013). [5] C. Giardinà, J. Kurchan, V. Lecomte, and J. Tailleur, J. Stat. Phy. 145, 787 (2011). [6] C. Giardinà, J. Kurchan, and L. Peliti, Phy.Rev.Lett.96, (2006). [7] J. Tailleur and J. Kurchan, Nat. Phy. 3, 203 (2007). [8] E. Guevara Hidalgo and V. Lecomte, J. Phy. A: Math. Theor. 49, (2016). [9] P. I. Hurtado and P. L. Garrido, J. Stat. Mech. (2009) P [10] M. Tchernookov and A. R. Dinner, J. Stat. Mech. (2010) P [11] A. Kundu, S. Sabhapandit, and A. Dhar, Phy. Rev. E 83, (2011). [12] T. Nemoto, F. Bouchet, R. L. Jack, and V. Lecomte, Phy. Rev. E 93, (2016). [13] T. Nemoto, E. Guevara Hidalgo, and V. Lecomte, Phy. Rev. E 95, (2017). [14] V. Lecomte and J. Tailleur, J. Stat. Mech. (2007) P [15] J. Tailleur and V. Lecomte, Modeling and Simulation of New Material, Proceeding of Modeling and Simulation of New Material: Tenth Granada Lecture, Granada (Spain), AIP Conf. Proc (AIP, Melville, NY, 2009), pp [16] H. Touchette, Phy. Rep. 478, 1 (2009). [17] J. P. Garrahan, R. L. Jack, V. Lecomte, E. Pitard, K. van Duijvendijk, and F. van Wijland, J. Phy. A 42, (2009). [18] T. E. Harri, Ann. Prob. 2, 969 (1974). [19] C. Bezuidenhout and G. Grimmett, Ann. Probab. 18, 1462 (1990). [20] V. Lecomte, C. Appert-Rolland, and F. van Wijland, J. Stat. Phy. 127, 51 (2007). [21] C. M. Rohwer, F. Angeletti, and H. Touchette, Phy. Rev. E 92, (2015). [22] A. B. Kolton, S. Butingorry, E. E. Ferrero, and A. Roo, J. Stat. Mech. (2013) P [23] P. T. Nyawo and H. Touchette, Europhy. Lett. 116, (2016). [24] T. Nemoto, R. L. Jack, and V. Lecomte, Phy.Rev.Lett.118, (2017)