Applied Probability

ISSN 1050-5164 (print) ISSN 2168-8737 (online) THE ANNALS of APPLIED PROBABILITY

AN OFFICIAL JOURNAL OF THE INSTITUTE OF MATHEMATICAL STATISTICS

Articles The length of the longest common subsequence of two independent mallows permutations KE JIN 1311 Determinantofsamplecorrelationmatrixwithapplication...... TIEFENG JIANG 1356 Annealed limit theorems for the Ising model on random regular graphs. . . VAN HAO CAN 1398 Approximating geodesics via random points...... ERIK DAVIS AND SUNDER SETHURAMAN 1446 Freidlin–Wentzell LDP in path space for McKean–Vlasov equations and the functional iterated logarithm law GONÇALO DOS REIS,WILLIAM SALKELD AND JULIAN TUGAUT 1487 A constrained Langevin approximation for chemical reaction networks SAUL C. LEITE AND RUTH J. WILLIAMS 1541 On a Wasserstein-type distance between solutions to stochastic differential equations JOCELYNE BION–NADAL AND DENIS TALAY 1609 Numerical method for FBSDEs of McKean–Vlasov type JEAN-FRANÇOIS CHASSAGNEUX,DAN CRISAN AND FRANÇOIS DELARUE 1640 Second-order BSDE under monotonicity condition and liquidation problem under uncertainty...... ALEXANDRE POPIER AND CHAO ZHOU 1685 Supermarket model on graphs AMARJIT BUDHIRAJA,DEBANKUR MUKHERJEE AND RUOYU WU 1740 Effective Berry–Esseen and concentration bounds for Markov chains with a spectral gap BENOÎT KLOECKNER 1778 The nested Kingman coalescent: Speed of coming down from inﬁnity AIRAM BLANCAS,TIM ROGERS,JASON SCHWEINSBERG AND ARNO SIRI-JÉGOUSSE 1808 Condensation in critical Cauchy Bienaymé–Galton–Watson trees IGOR KORTCHEMSKI AND LOÏC RICHIER 1837 Entropy-controlledLast-PassagePercolation...... QUENTIN BERGER AND NICCOLÒ TORRI 1878 The left-curtain martingale coupling in the presence of atoms DAV I D G. HOBSON AND DOMINYKAS NORGILAS 1904 Upper bounds for the function solution of the homogeneous 2D Boltzmann equation with hardpotential...... VLAD BALLY 1929

Vol. 29, No. 3—June 2019 THE ANNALS OF APPLIED PROBABILITY Vol. 29, No. 3, pp. 1311–1961 June 2019 INSTITUTE OF MATHEMATICAL STATISTICS

(Organized September 12, 1935)

The purpose of the Institute is to foster the development and dissemination of the theory and applications of statistics and probability.

IMS OFFICERS

President: Xiao-Li Meng, Department of Statistics, Harvard University, Cambridge, Massachusetts 02138-2901, USA

President-Elect: Susan Murphy, Department of Statistics, Harvard University, Cambridge, Massachusetts 02138- 2901, USA

Past President: Alison Etheridge, Department of Statistics, University of Oxford, Oxford, OX1 3LB, United King- dom

Executive Secretary: Edsel Peña, Department of Statistics, University of South Carolina, Columbia, South Car- olina 29208-001, USA

Treasurer: Zhengjun Zhang, Department of Statistics, University of Wisconsin, Madison, Wisconsin 53706-1510, USA

Program Secretary: Ming Yuan, Department of Statistics, Columbia University, New York, NY 10027-5927, USA

IMS EDITORS

The Annals of Statistics. Editors: Richard J. Samworth, Statistical Laboratory, Centre for Mathematical Sciences, University of Cambridge, Cambridge, CB3 0WB, UK. Ming Yuan, Department of Statistics, Columbia Uni- versity, New York, NY 10027, USA

The Annals of Applied Statistics. Editor-in-Chief : Karen Kafadar, Department of Statistics, University of Virginia, Heidelberg Institute for Theoretical Studies, Charlottesville, VA 22904-4135, USA

The Annals of Probability. Editor: Amir Dembo, Department of Statistics and Department of Mathematics, Stan- ford University, Stanford, California 94305, USA

The Annals of Applied Probability. Editors: François Delarue, Laboratoire J. A. Dieudonné, Université de Nice Sophia-Antipolis, France-06108 Nice Cedex 2. Peter Friz, Institut für Mathematik, Technische Universität Berlin, 10623 Berlin, Germany and Weierstrass-Institut für Angewandte Analysis und Stochastik, 10117 Berlin, Germany

Statistical Science. Editor: Cun-Hui Zhang, Department of Statistics, Rutgers University, Piscataway, New Jersey 08854, USA

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The Annals of Applied Probability [ISSN 1050-5164 (print); ISSN 2168-8737 (online)], Volume 29, Number 3, June 2019. Published bimonthly by the Institute of Mathematical Statistics, 3163 Somerset Drive, Cleveland, Ohio 44122, USA. Periodicals postage paid at Cleveland, Ohio, and at additional mailing ofﬁces.

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Copyright © 2019 by the Institute of Mathematical Statistics Printed in the United States of America The Annals of Applied Probability 2019, Vol. 29, No. 3, 1311–1355 https://doi.org/10.1214/17-AAP1351 © Institute of Mathematical Statistics, 2019

THE LENGTH OF THE LONGEST COMMON SUBSEQUENCE OF TWO INDEPENDENT MALLOWS PERMUTATIONS1

BY KE JIN University of Delaware

The Mallows measure is a probability measure on Sn where the probability of a permutation π is proportional to ql(π) with q>0 being a parameter and l(π) the number of inversions in π. We prove a weak law of large numbers for the length of the longest common subsequences of two independent permutations drawn from the Mallows measure, when q is a function of n and n(1 − q) has limit in R as n →∞.

REFERENCES

ABELLO, J. (1991). The weak Bruhat order of S consistent sets, and Catalan numbers. SIAM J. Discrete Math. 4 1–16. MR1090284 BAIK,J.,DEIFT,P.andJOHANSSON, K. (1999). On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12 1119–1178. BHATNAGAR,N.andPELED, R. (2015). Lengths of monotone subsequences in a Mallows permutation. Probab. Theory Related Fields 161 719–780. CHVÁTAL,V.andSANKOFF, D. (1975). Longest common subsequences of two random sequences. J. Appl. Probab. 12 306–315. MR0405531 CRITCHLOW, D. E. (1985). Metric Methods for Analyzing Partially Ranked Data. Lecture Notes in Statistics 34. Springer, Berlin. MR0818986 DANCÍKˇ , V. (1994). Expected length of longest common subsequences. Ph.D. dissertation, Univ. Warwick. DANCÍKˇ ,V.andPATERSON, M. (1995). Upper bounds for the expected length of a longest common subsequence of two binary sequences. Random Structures Algorithms 6 449–458. MR1368846 DEKEN, J. G. (1979). Some limit results for longest common subsequences. Discrete Math. 26 17– 31. DEUSCHEL,J.-D.andZEITOUNI, O. (1995). Limiting curves for i.i.d. records. Ann. Probab. 23 852–878. MR1334175 FLIGNER,M.A.andVERDUCCI, J. S. (1993). Probability Models and Statistical Analyses for Ranking Data. Lecture Notes in Statistics 80. Springer, Berlin. HAMMERSLEY, J. (1972). A few seedlings of research. In Proc. of the Sixth Berkeley Symp. Math. Statist. and Probability, Vol. 1 345–394. Univ. California Press, Berkeley, CA. MR0405665 HOPPEN,C.,KOHAYAKAWA,Y.,MOREIRA,C.G.,RÁTH,B.andSAMPAIO, R. M. (2013). Limits of permutation sequences. J. Combin. Theory Ser. B 103 93–113. HOUDRÉ,C.andISLAK¸ , Ü. (2014). A central limit theorem for the length of the longest common subsequence in random words. Available at: arXiv:1408.1559. JIN, K. (2017). The limit of the empirical measure of the product of two independent Mallows permutations. Available at: arXiv:1702.00140.

MSC2010 subject classiﬁcations. 60F05, 60B15, 05A05. Key words and phrases. Longest common subsequence, longest increasing subsequence, Mal- lows measure. KENYON,R.,KRAL,D.,RADIN,C.andWINKLER, P. (2015). Permutations with ﬁxed pattern densities. Available at: arXiv:1506.02340. KEROV,S.andVERSHIK, A. (1977). Asymptotic behavior of the Plancherel measure of the symmetric group and the limit form of Young tableaux. Sov. Math. Dokl. 18 527–531. LINDVALL, T. et al. (1999). On Strassen’s theorem on stochastic domination. Electron. Commun. Probab. 4 51–59. LOGAN,B.F.andSHEPP, L. A. (1977). A variational problem for random Young tableaux. Adv. Math. 26 206–222. LUEKER, G. S. (2009). Improved bounds on the average length of longest common subsequences. J. ACM 56 17. MALLOWS, C. L. (1957). Non-null ranking models. I. Biometrika 44 114–130. MR0087267 MARDEN, J. I. (1995). Analyzing and Modeling Rank Data. Monographs on Statistics and Applied Probability 64. Chapman & Hall, London. MR1346107 MUELLER,C.andSTARR, S. (2013). The length of the longest increasing subsequence of a random Mallows permutation. J. Theoret. Probab. 26 514–540. MR3055815 MUKHERJEE, S. et al. (2016). Fixed points and cycle structure of random permutations. Electron. J. Probab. 21. PEVZNER, P. (2000). Computational Molecular Biology: An Algorithmic Approach. MIT Press, Cambridge, MA. STARR, S. (2009). Thermodynamic limit for the Mallows model on S_n. Available at: arXiv:0904.0696. The Annals of Applied Probability 2019, Vol. 29, No. 3, 1356–1397 https://doi.org/10.1214/17-AAP1362 © Institute of Mathematical Statistics, 2019

DETERMINANT OF SAMPLE CORRELATION MATRIX WITH APPLICATION

BY TIEFENG JIANG1 University of Minnesota

Let x1,...,xn be independent random vectors of a common p-dimensional normal distribution with population correlation matrix Rn.Thesample ˆ correlation matrix Rn = (rîj )p×p is generated from x1,...,xn such that rîj is the Pearson correlation coefficient between the ith column and the jth ˆ column of the data matrix (x1,...,xn) . The matrix Rn is a popular object in multivariate analysis and it has many connections to other problems. We derive a central limit theorem (CLT) for the logarithm of the determinant of ˆ Rn for a big class of Rn. The expressions of mean and the variance in the CLT are not obvious, and they are not known before. In particular, the CLT holds if p/n has a nonzero limit and the smallest eigenvalue of Rn is larger than ˆ 1/2. Besides, a formula of the moments of |Rn| and a new method of showing weak convergence are introduced. We apply the CLT to a high-dimensional statistical test.

REFERENCES

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MSC2010 subject classifications. 60B20, 60F05. Key words and phrases. Central limit theorem, sample correlation matrix, smallest eigenvalue, multivariate normal distribution, moment generating function. DONG,Z.,JIANG,T.andLI, D. (2012). Circular law and arc law for truncation of random unitary matrix. J. Math. Phys. 53 Article ID 013301. MR2919538 EATON, M. L. (1983). Multivariate Statistics: A Vector Space Approach. Wiley, New York. MR0716321 HANSON,D.L.andWRIGHT, F. T. (1971). A bound on tail probabilities for quadratic forms in independent random variables. Ann. Math. Stat. 42 1079–1083. MR0279864 HORN,R.A.andJOHNSON, C. R. (1985). Matrix Analysis. Cambridge Univ. Press, Cambridge. MR0832183 JIANG, T. (2004a). The asymptotic distributions of the largest entries of sample correlation matrices. Ann. Appl. Probab. 14 865–880. MR2052906 JIANG, T. (2004b). The limiting distributions of eigenvalues of sample correlation matrices. Sankhya¯ 66 35–48. JIANG,T.andQI, Y. (2015). Likelihood ratio tests for high-dimensional normal distributions. Scand. J. Stat. 42 988–1009. MR3426306 JIANG,T.andYANG, F. (2013). Central limit theorems for classical likelihood ratio tests for high- dimensional normal distributions. Ann. Statist. 41 2029–2074. MR3127857 LI,D.,LIU,W.andROSALSKY, A. (2010). Necessary and sufficient conditions for the asymptotic distribution of the largest entry of a sample correlation matrix. Probab. Theory Related Fields 148 5–35. MR2653220 LI,D.andROSALSKY, A. (2006). Some strong limit theorems for the largest entries of sample correlation matrices. Ann. Appl. Probab. 16 423–447. MR2209348 MORRISON, D. F. (2004). Multivariate Statistical Methods, 4th ed. Duxbury Press, Pacific Grove, CA. MUIRHEAD, R. J. (1982). Aspects of Multivariate Statistical Theory. Wiley, New York. NGUYEN,H.H.andVU, V. (2014). Random matrices: Law of the determinant. Ann. Probab. 42 146–167. MR3161483 RUDELSON,M.andVERSHYNIN, R. (2013). Hanson–Wright inequality and sub-Gaussian concentration. Electron. Commun. Probab. 18 Article ID 82. MR3125258 SMALE, S. (2000). Mathematical problems for the next century. In Mathematics: Frontiers and Perspectives (V. Arnold, M. Atiyah, P. Lax and B. Mazur, eds.) 271–294. Amer. Math. Soc., Providence, RI. MR1754783 TAO,T.andVU, V. (2012). A central limit theorem for the determinant of a Wigner matrix. Adv. Math. 231 74–101. MR2935384 WILKS, S. S. (1932). Certain generalizations in the analysis of variance. Biometrika 24 471–494. YURINSKII˘, V. V. (1976). Exponential inequalities for sums of random vectors. J. Multivariate Anal. 6 473–499. MR0428401 ZHOU, W. (2007). Asymptotic distribution of the largest off-diagonal entry of correlation matrices. Trans. Amer. Math. Soc. 359 5345–5363. MR2327033 The Annals of Applied Probability 2019, Vol. 29, No. 3, 1398–1445 https://doi.org/10.1214/17-AAP1377 © Institute of Mathematical Statistics, 2019

ANNEALED LIMIT THEOREMS FOR THE ISING MODEL ON RANDOM REGULAR GRAPHS

BY VAN HAO CAN Vietnam Academy of Science and Technology

In a recent paper, Giardinà et al. [ALEA Lat. Am. J. Probab. Math. Stat. 13 (2016) 121–161] have proved a law of large number and a central limit theorem with respect to the annealed measure for the magnetization of the Ising model on some random graphs, including the random 2-regular graph. In this paper, we present a new proof of their results which applies to all random regular graphs. In addition, we prove the existence of annealed pressure in the case of conﬁguration model random graphs.

REFERENCES

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APPROXIMATING GEODESICS VIA RANDOM POINTS1

BY ERIK DAVIS AND SUNDER SETHURAMAN University of Arizona

⊂ Rd Given a cost functional F on paths γ in a domain D , in the form = 1 ˙ F(γ) 0 f(γ(t),γ(t))dt, it is of interest to approximate its minimum cost and geodesic paths. Let X1,...,Xn be points drawn independently from D according to a distribution with a density. Form a random geometric graph on the points where Xi and Xj are connected when 0 < |Xi − Xj | <ε,andthe length scale ε = εn vanishes at a suitable rate. For a general class of functionals F , associated to Finsler and other distances on D, using a probabilistic form of Gamma convergence, we show that the minimum costs and geodesic paths, with respect to types of approximating discrete cost functionals, built from the random geometric graph, con- verge almost surely in various senses to those corresponding to the continuum cost F , as the number of sample points diverges. In particular, the geodesic path convergence shown appears to be among the ﬁrst results of its kind.

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FREIDLIN–WENTZELL LDP IN PATH SPACE FOR MCKEAN–VLASOV EQUATIONS AND THE FUNCTIONAL ITERATED LOGARITHM LAW

∗ ∗ BY GONÇALO DOS REIS ,1,WILLIAM SALKELD AND JULIAN TUGAUT† University of Edinburgh∗ and Université Jean Monnet†

We show two Freidlin–Wentzell-type Large Deviations Principles (LDP) in path space topologies (uniform and Hölder) for the solution process of McKean–Vlasov Stochastic Differential Equations (MV-SDEs) using techniques which directly address the presence of the law in the coefﬁcients and altogether avoiding decoupling arguments or limits of particle systems. We provide existence and uniqueness results along with several properties for a class of MV-SDEs having random coefﬁcients and drifts of superlinear growth. As an application of our results, we establish a functional Strassen-type result (law of iterated logarithm) for the solution process of a MV-SDE.

REFERENCES

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MSC2010 subject classifications. Primary 60F10; secondary 60G07. Key words and phrases. McKean–Vlasov equations, large deviations principle, path-space, Hölder topologies, superlinear growth, functional Strassen law. [9] CARMONA,R.andDELARUE, F. (2018). Probabilistic Theory of Mean Field Games with Applications. II. Probability Theory and Stochastic Modelling 84. Springer, Cham. MR3753660 [10] CATTIAUX,P.,GUILLIN,A.andMALRIEU, F. (2008). Probabilistic approach for granular media equations in the non-uniformly convex case. Probab. Theory Related Fields 140 19–40. MR2357669 [11] COLLET,F.,DAI PRA,P.andSARTORI, E. (2010). A simple mean field model for social interactions: Dynamics, fluctuations, criticality. J. Stat. Phys. 139 820–858. MR2639892 [12] CRISAN,D.andMCMURRAY, E. (2018). Smoothing properties of McKean–Vlasov SDEs. Probab. Theory Related Fields 171 97–148. MR3800831 [13] CRISAN,D.andXIONG, J. (2010). Approximate McKean–Vlasov representations for a class of SPDEs. 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A CONSTRAINED LANGEVIN APPROXIMATION FOR CHEMICAL REACTION NETWORKS

BY SAUL C. LEITE1 AND RUTH J. WILLIAMS2 Universidade Federal do ABC and University of California, San Diego

Continuous-time Markov chain models are often used to describe the stochastic dynamics of networks of reacting chemical species, especially in the growing field of systems biology. These Markov chain models are often studied by simulating sample paths in order to generate Monte-Carlo estimates. However, discrete-event stochastic simulation of these models rapidly becomes computationally intensive. Consequently, more tractable diffusion approximations are commonly used in numerical computation, even for modest-sized networks. However, existing approximations either do not respect the constraint that chemical concentrations are never negative (linear noise approximation) or are typically only valid until the concentration of some chemical species first becomes zero (Langevin approximation). In this paper, we propose an approximation for such Markov chains via reflected diffusion processes that respect the fact that concentrations of chemical species are never negative. We call this a constrained Langevin approximation because it behaves like the Langevin approximation in the interior of the positive orthant, to which it is constrained by instantaneous reflection at the boundary of the orthant. An additional advantage of our approximation is that it can be written down immediately from the chemical reactions. This contrasts with the linear noise approximation, which involves a two- stage procedure—first solve a deterministic reaction rate ordinary differential equation, followed by a stochastic differential equation for fluctuations around those solutions. Our approximation also captures the interaction of nonlinearities in the reaction rate function with the driving noise. In simula- tions, we have found the computation time for our approximation to be at least comparable to, and often better than, that for the linear noise approximation. Under mild assumptions, we first prove that our proposed approximation is well defined for all time. Then we prove that it can be obtained as the weak limit of a sequence of jump-diffusion processes that behave like the Langevin approximation in the interior of the positive orthant and like a rescaled version of the Markov chain on the boundary of the orthant. For this limit theorem, we adapt an invariance principle for reflected diffusions, due to Kang and Williams [Ann. Appl. Probab. 17 (2007) 741–779], and modify a result on pathwise uniqueness for reflected diffusions due to Dupuis and Ishii [Ann. Probab. 21 (1993) 554–580]. Some numerical examples illustrate the advan-

MSC2010 subject classiﬁcations. Primary 60J28, 60J60, 60F17, 92C45; secondary 60H10, 92C40. Key words and phrases. Diffusion approximation, Langevin approximation, chemical reaction networks, stochastic differential equation with reﬂection, scaling limit, linear noise approximation, density dependent Markov chains, jump-diffusion. tages of our approximation over direct simulation of the Markov chain or use of the linear noise approximation.

REFERENCES

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ON A WASSERSTEIN-TYPE DISTANCE BETWEEN SOLUTIONS TO STOCHASTIC DIFFERENTIAL EQUATIONS

BY JOCELYNE BION–NADAL1 AND DENIS TALAY École Polytechnique and INRIA

In this paper, we introduce a Wasserstein-type distance on the set of the probability distributions of strong solutions to stochastic differential equations. This new distance is defined by restricting the set of possible coupling measures. We prove that it may also be defined by means of the value function of a stochastic control problem whose Hamilton–Jacobi–Bellman equation has a smooth solution, which allows one to deduce a priori estimates or to obtain numerical evaluations. We exhibit an optimal coupling measure and characterize it as a weak solution to an explicit stochastic differential equation, and we finally describe procedures to approximate this optimal coupling measure. A notable application concerns the following modeling issue: given an exact diffusion model, how to select a simplified diffusion model within a class of admissible models under the constraint that the probability distribution of the exact model is preserved as much as possible?

REFERENCES

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NUMERICAL METHOD FOR FBSDES OF MCKEAN–VLASOV TYPE

BY JEAN-FRANÇOIS CHASSAGNEUX,DAN CRISAN AND FRANÇOIS DELARUE Université Paris Diderot, Imperial College London and Université de Nice Sophia-Antipolis

This paper is dedicated to the presentation and the analysis of a numerical scheme for forward–backward SDEs of the McKean–Vlasov type, or equivalently for solutions to PDEs on the Wasserstein space. Because of the mean field structure of the equation, earlier methods for classical forward– backward systems fail. The scheme is based on a variation of the method of continuation. The principle is to implement recursively local Picard iterations on small time intervals. We establish a bound for the rate of convergence under the assumption that the decoupling field of the forward–backward SDE (or equivalently the solution of the PDE) satisfies mild regularity conditions. We also provide numerical illustrations.

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SECOND-ORDER BSDE UNDER MONOTONICITY CONDITION AND LIQUIDATION PROBLEM UNDER UNCERTAINTY

BY ALEXANDRE POPIER1 AND CHAO ZHOU2 Le Mans Université and National University of Singapore

In this work, we investigate an optimal liquidation problem under Knigh- tian uncertainty. We obtain the value function and an optimal control characterised by the solution of a second-order BSDE with monotone generator and with a singular terminal condition.

REFERENCES

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SUPERMARKET MODEL ON GRAPHS

BY AMARJIT BUDHIRAJA1,DEBANKUR MUKHERJEE2 AND RUOYU WU University of North Carolina, Chapel Hill, Brown University and University of Michigan

We consider a variation of the supermarket model in which the servers can communicate with their neighbors and where the neighborhood relation- ships are described in terms of a suitable graph. Tasks with unit-exponential service time distributions arrive at each vertex as independent Poisson processes with rate λ, and each task is irrevocably assigned to the shortest queue among the one it first appears and its d − 1 randomly selected neighbors. This model has been extensively studied when the underlying graph is a clique in which case it reduces to the well-known power-of-d scheme. In particular, results of Mitzenmacher (1996) and Vvedenskaya et al. (1996) show that as the size of the clique gets large, the occupancy process associated with the queue-lengths at the various servers converges to a deterministic limit described by an infinite system of ordinary differential equations (ODE). In this work, we consider settings where the underlying graph need not be a clique and is allowed to be suitably sparse. We show that if the minimum degree approaches infinity (however slowly) as the number of servers N approaches infinity, and the ratio between the maximum degree and the minimum degree in each connected component approaches 1 uniformly, the occupancy process converges to the same system of ODE as the classical supermarket model. In particular, the asymptotic behavior of the occupancy process is in- sensitive to the precise network topology. We also study the case where the graph sequence is random, with the Nth graph given as an Erdos–Rényi˝ random graph on N vertices with average degree c(N). Annealed convergence of the occupancy process to the same deterministic limit is established under the condition c(N) →∞, and under a stronger condition c(N)/ln N →∞, convergence (in probability) is shown for almost every realization of the random graph.

REFERENCES

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The Annals of Applied Probability 2019, Vol. 29, No. 3, 1778–1807 https://doi.org/10.1214/18-AAP1438 © Institute of Mathematical Statistics, 2019

EFFECTIVE BERRY–ESSEEN AND CONCENTRATION BOUNDS FOR MARKOV CHAINS WITH A SPECTRAL GAP

BY BENOÎT KLOECKNER Université Paris-Est Applying quantitative perturbation theory for linear operators, we prove nonasymptotic bounds for Markov chains whose transition kernel has a spectral gap in an arbitrary Banach algebra of functions X . The main results are concentration inequalities and Berry–Esseen bounds, obtained assuming nei- ther reversibility nor “warm start” hypothesis: the law of the ﬁrst term of the chain can be arbitrary. The spectral gap hypothesis is basically a uniform X - ergodicity hypothesis, and when X consist in regular functions this is weaker than uniform ergodicity. We show on a few examples how the ﬂexibility in the choice of function space can be used. The constants are completely explicit and reasonable enough to make the results usable in practice, notably in MCMC methods.

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MSC2010 subject classiﬁcations. Primary 65C05; secondary 60J22, 62E17. Key words and phrases. Concentration inequalities, Berry–Esseen bounds, Markov chains, spectral gap property, Markov chain Monte-Carlo method. [12] FELLER, W. (1966). An Introduction to Probability Theory and Its Applications. Vol. II. Wiley, New York. MR0210154 [13] GIULIETTI,P.,KLOECKNER,B.,LOPES,A.O.andMARCON, D. (2018). The calculus of thermodynamical formalism. J. Eur. Math. Soc.(JEMS) 20 2357–2412. MR3852182 [14] GLYNN,P.W.andORMONEIT, D. (2002). Hoeffding’s inequality for uniformly ergodic Markov chains. Statist. Probab. Lett. 56 143–146. MR1881167 [15] GÓMEZ,D.M.andDARTNELL, P. (2012). Simple Monte Carlo integration with respect to Bernoulli convolutions. Appl. Math. 57 617–626. MR3010240 [16] GOUËZEL, S. (2015). Limit theorems in dynamical systems using the spectral method. In Hy- perbolic Dynamics, Fluctuations and Large Deviations. Proc. Sympos. Pure Math. 89 161–193. Amer. Math. Soc., Providence, RI. MR3309098 [17] HENNION,H.andHERVÉ, L. (2001). Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness. Lecture Notes in Math. 1766. Springer, Berlin. MR1862393 [18] JOULIN,A.andOLLIVIER, Y. (2010). Curvature, concentration and error estimates for Markov chain Monte Carlo. Ann. Probab. 38 2418–2442. MR2683634 [19] KELLER,G.andLIVERANI, C. (1999). Stability of the spectrum for transfer operators. Ann. Sc. Norm. Super. Pisa, Cl. Sci.(4)28 141–152. MR1679080 [20] KLOECKNER, B. R. (2017). Effective perturbation theory for simple isolated eigenvalues of linear operators. J. Operator Theory To appear. Available at arXiv:1703.09425. [21] KLOECKNER, B. R. (2017). Effective high-temperature estimates for intermittent maps. Er- godic Theory Dynam. Systems To appear. Available at arXiv:1704.00586. [22] KLOECKNER, B. R. (2017). An optimal transportation approach to the decay of correlations for non-uniformly expanding maps. Ergodic Theory Dynam. Systems.Toappear.Availableat arXiv:1711.08052. [23] KLOECKNER, B. R. (2018). Toy examples for effective concentration bounds. [24] KONTOYIANNIS,I.,LASTRAS-MONTANO,L.A.andMEYN, S. P. (2005). Relative entropy and exponential deviation bounds for general Markov chains. In International Symposium on Information Theory, 2005 1563–1567. IEEE Press, New York. [25] KONTOYIANNIS,I.andMEYN, S. P. (2012). Geometric ergodicity and the spectral gap of non-reversible Markov chains. Probab. Theory Related Fields 154 327–339. MR2981426 [26] LEZAUD, P. (1998). Chernoff-type bound for ﬁnite Markov chains. Ann. Appl. Probab. 8 849– 867. MR1627795 [27] LEZAUD, P. (2001). Chernoff and Berry–Esséen inequalities for Markov processes. ESAIM Probab. Stat. 5 183–201. MR1875670 [28] LIVERANI, C. (2001). Rigorous numerical investigation of the statistical properties of piecewise expanding maps. A feasibility study. Nonlinearity 14 463–490. MR1830903 [29] NAGAEV, S. V. (1957). Some limit theorems for stationary Markov chains. Teor. Veroyatn. Primen. 2 389–416. MR0094846 [30] NAGAEV, S. V. (1961). More exact limit theorems for homogeneous Markov chains. Teor. Veroyatn. Primen. 6 67–86. MR0131291 [31] PAULIN, D. (2015). Concentration inequalities for Markov chains by Marton couplings and spectral methods. Electron. J. Probab. 20 no. 79, 32. MR3383563 [32] PAULIN, D. (2016). Mixing and concentration by Ricci curvature. J. Funct. Anal. 270 1623– 1662. MR3452712 [33] PERES,Y.,SCHLAG,W.andSOLOMYAK, B. (2000). Sixty years of Bernoulli convolutions. In Fractal Geometry and Stochastics, II (Greifswald/Koserow, 1998). Progress in Probability 46 39–65. Birkhäuser, Basel. MR1785620 [34] ROBERTS,G.O.andROSENTHAL, J. S. (2004). General state space Markov chains and MCMC algorithms. Probab. Surv. 1 20–71. MR2095565 [35] RUELLE, D. (2004). Thermodynamic Formalism: The Mathematical Structures of Equilibrium Statistical Mechanics, 2nd ed. Cambridge Mathematical Library. Cambridge Univ. Press, Cambridge. MR2129258 [36] SOLOMYAK, B. (1995). On the random series ±λn (an Erdos˝ problem). Ann. of Math.(2) 142 611–625. MR1356783 [37] TYURIN, I. S. (2011). Improvement of the remainder in the Lyapunov theorem. Teor. Veroyatn. Primen. 56 808–811. MR3137072 [38] WATANABE,S.andHAYASHI, M. (2017). Finite-length analysis on tail probability for Markov chain and application to simple hypothesis testing. Ann. Appl. Probab. 27 811–845. MR3655854 The Annals of Applied Probability 2019, Vol. 29, No. 3, 1808–1836 https://doi.org/10.1214/18-AAP1440 © Institute of Mathematical Statistics, 2019

THE NESTED KINGMAN COALESCENT: SPEED OF COMING DOWN FROM INFINITY

∗ BY AIRAM BLANCAS ,1,TIM ROGERS†,2,JASON SCHWEINSBERG‡,3 AND ARNO SIRI-JÉGOUSSE§,4 Goethe Universität Frankfurt∗, University of Bath†, University of California, San Diego‡ and Universidad Nacional Autónoma de México§

The nested Kingman coalescent describes the ancestral tree of a population undergoing neutral evolution at the level of individuals and at the level of species, simultaneously. We study the speed at which the number of lineages descends from inﬁnity in this hierarchical coalescent process and prove the existence of an early-time phase during which the number of lineages at time t decays as 2γ/ct2,wherec is the ratio of the coalescence rates at the individ- ual and species levels, and the constant γ ≈ 3.45 is derived from a recursive distributional equation for the number of lineages contained within a species at a typical time.

REFERENCES

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MSC2010 subject classiﬁcations. Primary 60J25; secondary 60J80, 92D15, 92D25. Key words and phrases. Kingman’s coalescent, nested coalescent, gene tree, species tree, coming down from inﬁnity, recursive distributional equation. [11] EWENS, W. J. (1972). The sampling theory of selectively neutral alleles. Theor. Popul. Biol. 3 87–112; erratum, ibid. 3 (1972), 240; erratum, ibid. 3 (1972), 376. MR0325177 [12] GIVENS,C.R.andSHORTT, R. M. (1984). A class of Wasserstein metrics for probability distributions. Michigan Math. J. 31 231–240. MR0752258 [13] HARVEY,P.H.,MAY,R.M.andNEE, S. (1994). Phylogenies without fossils. Evolution 48 523–529. [14] KANTOROVICˇ ,L.V.andRUBINŠTEIN˘ , G. Š. (1958). On a space of completely additive functions. Vestnik Leningrad Univ. Math. 13 52–59. MR0102006 [15] KINGMAN, J. F. C. (1982). The coalescent. Stochastic Process. Appl. 13 235–248. MR0671034 [16] LAMBERT,A.andSCHERTZER, E. (2018). Coagulation-transport equations and the nested coalescents. Available at arXiv:1807.09153. [17] MADDISON, W. P. (1997). Gene trees in species trees. Syst. Biol. 46 523–536. [18] MÖHLE, M. (2000). Total variation distances and rates of convergence for ancestral coalescent processes in exchangeable population models. Adv. in Appl. Probab. 32 983–993. MR1808909 [19] MOOERS,A.O.andHEARD, S. B. (1997). Inferring evolutionary process from phylogenetic tree shape. Q. Rev. Biol. 72 31–54. [20] MORAN, P. A. P. (1958). Random processes in genetics. Proc. Camb. Philos. Soc. 54 60–71. MR0127989 [21] MORGAN,M.J.,BASS,D.,BIK,H.,BIRKY,C.W.,BLAXTER,M.,CRISP,M.D., DERYCKE,S.,FITCH,D.,FONTANETO, D. et al. (2014). A critique of Rossberg et al.: Noise obscures the genetic signal of meiobiotal ecospecies in ecogenomic datasets. Proc. Royal Soc., Biol. Sci. 281 20133076. [22] ROSSBERG,A.G.,ROGERS,T.andMCKANE, A. J. (2013). Are there species smaller than 1 mm? Proc. Royal Soc. Biol. Sci. 280 20131248. [23] ROSSBERG,A.G.,ROGERS,T.andMCKANE, A. J. (2014). Current noise-removal methods can create false signals in ecogenomic data. Proc. Royal Soc. Biol. Sci. 281 20140191. [24] SZÖLLOSI˝ ,G.J.,TANNIER,E.,DAUBIN,V.andBOUSSAU, B. (2014). The inference of gene trees with species trees. Syst. Biol. 64 e42–e62. The Annals of Applied Probability 2019, Vol. 29, No. 3, 1837–1877 https://doi.org/10.1214/18-AAP1447 © Institute of Mathematical Statistics, 2019

CONDENSATION IN CRITICAL CAUCHY BIENAYMÉ–GALTON–WATSON TREES1

∗ BY IGOR KORTCHEMSKI ,†,2 AND LOÏC RICHIER† CNRS∗ & CMAP, École polytechnique†

We are interested in the structure of large Bienaymé–Galton–Watson random trees whose offspring distribution is critical and falls within the domain of attraction of a stable law of index α = 1. In stark contrast to the case α ∈ (1, 2], we show that a condensation phenomenon occurs: in such trees, one vertex with macroscopic degree emerges (see Figure 1). To this end, we establish limit theorems for centered downwards skip-free random walks whose steps are in the domain of attraction of a Cauchy distribution, when conditioned on a late entrance in the negative real line. These results are of independent interest. As an application, we study the geometry of the boundary of random planar maps in a speciﬁc regime (called nongeneric of parameter 3/2). This supports the conjecture that faces in Le Gall and Mier- mont’s 3/2-stable maps are self-avoiding.

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MR0408020 The Annals of Applied Probability 2019, Vol. 29, No. 3, 1878–1903 https://doi.org/10.1214/18-AAP1448 © Institute of Mathematical Statistics, 2019

ENTROPY-CONTROLLED LAST-PASSAGE PERCOLATION1

BY QUENTIN BERGER2 AND NICCOLÒ TORRI3 Sorbonne Université We introduce a natural generalization of Hammersley’s Last-Passage Percolation (LPP) called Entropy-controlled Last-Passage Percolation (E- LPP), where points can be collected by paths with a global (path-entropy) constraint which takes into account the whole structure of the path, instead of a local (1-Lipschitz) constraint as in Hammersley’s LPP. Our main result is to prove quantitative tail estimates on the maximal number of points that can be collected by a path with entropy bounded by a prescribed constant. The E-LPP turns out to be a key ingredient in the context of the directed polymer model when the environment is heavy-tailed, which we consider in (Berger and Torri (2018)). We give applications in this context, which are essentials tools in (Berger and Torri (2018)): we show that the limiting variational problem conjectured in (Ann. Probab. 44 (2016) 4006–4048), Conjecture 1.7, is ﬁnite, and we prove that the discrete variational problem converges to the continuous one, generalizing techniques used in (Comm. Pure Appl. Math. 64 (2011) 183–204; Probab. Theory Related Fields 137 (2007) 227–275).

REFERENCES

[1] ALBERTS,T.,KHANIN,K.andQUASTEL, J. (2014). The intermediate disorder regime for directed polymers in dimension 1 + 1. Ann. Probab. 42 1212–1256. MR3189070 [2] AUFFINGER,A.andLOUIDOR, O. (2011). Directed polymers in a random environment with heavy tails. Comm. Pure Appl. Math. 64 183–204. MR2766526 [3] BAIK,J.,DEIFT,P.andJOHANSSON, K. (1999). On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12 1119– 1178. MR1682248 [4] BERGER,Q.andTORRI, N. (2018). Directed polymers in heavy-tail random environment. Available at arXiv:1802.03355. [5] BERGER,Q.andTORRI, N. (2018). Beyond Hammersley’s Last-Passage Percolation: A dis- cussion on possible new local and global constraints. Available at ArXiv:1802.04046. [6] BINGHAM,N.H.,GOLDIE,C.M.andTEUGELS, J. L. (1989). Regular Variation. Ency- clopedia of Mathematics and Its Applications 27. Cambridge Univ. Press, Cambridge. MR1015093 [7] COMETS, F. (2016). Directed Polymers in Random Environments. Ecole d’Eté de probabilités de Saint-Flour 2175. Springer, Cham. [8] COMETS,F.,SHIGA,T.andYOSHIDA, N. (2004). Probabilistic analysis of directed polymers in a random environment: A review. In Stochastic Analysis on Large Scale Interacting Systems. Adv. Stud. Pure Math. 39 115–142. Math. Soc. Japan, Tokyo. MR2073332 [9] DEN HOLLANDER, F. (2007). Random Polymers. Ecole d’Eté de probabilités de Saint-Flour 1974. Springer, Berlin.

MSC2010 subject classiﬁcations. Primary 60K35; secondary 60K37, 60F05. Key words and phrases. Last-passage percolation, heavy-tail distributions, path entropy. [10] DEY,P.S.andZYGOURAS, N. (2016). High temperature limits for (1 + 1)-dimensional directed polymer with heavy-tailed disorder. Ann. Probab. 44 4006–4048. MR3572330 [11] HAMBLY,B.andMARTIN, J. B. (2007). Heavy tails in last-passage percolation. Probab. The- ory Related Fields 137 227–275. MR2278457 [12] HAMMERSLEY, J. M. (1972). A few seedlings of research. In Proc. Sixth Berkeley Symp. Math. Statist. and Probab. 345–394. Univ. California Press, Berkeley, CA. MR0405665 [13] LOGAN,B.F.andSHEPP, L. A. (1977). A variational problem for random Young tableaux. Adv. Math. 26 206–222. MR1417317 [14] RESNICK, S. I. (2008). Extreme Values, Regular Variation and Point Processes. Series in Op- erations Research and Financial Engineering. Springer, New York. MR2364939 [15] STONE, C. (1967). On local and ratio limit theorems. In Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66), Vol. II: Contributions to Probability Theory, Part 2 217–224. Univ. California Press, Berkeley, CA. MR0222939 [16] ULAM, S. M. (1961). Monte Carlo calculations in problems of mathematical physics. In Mod- ern Mathematics for the Engineer: Second Series 261–281. McGraw-Hill, New York. MR0129165 [17] VERSHIK,A.M.andKEROV, S. V. (1977). Asymptotics of the plancherel measure of the symmetric group and the limiting form of Young tables. Sov. Math., Dokl. 18 527–531. The Annals of Applied Probability 2019, Vol. 29, No. 3, 1904–1928 https://doi.org/10.1214/18-AAP1450 © Institute of Mathematical Statistics, 2019

THE LEFT-CURTAIN MARTINGALE COUPLING IN THE PRESENCE OF ATOMS

BY DAV I D G. HOBSON AND DOMINYKAS NORGILAS University of Warwick

Beiglböck and Juillet (Ann. Probab. 44 (2016) 42–106) introduced the left-curtain martingale coupling of probability measures μ and ν, and proved that, when the initial law μ is continuous, it is supported by the graphs of two functions. We extend the later result by constructing the generalised left- curtain martingale coupling and show that for an arbitrary starting law μ it is characterised by two appropriately deﬁned lower and upper functions. As an application of this result, we derive the model-independent upper bound of an American put option. This extends recent results of Hobson and Norgilas (2017) on the atom-free case.

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UPPER BOUNDS FOR THE FUNCTION SOLUTION OF THE HOMOGENEOUS 2D BOLTZMANN EQUATION WITH HARD POTENTIAL

BY VLAD BALLY Université Paris-Est

We deal with ft (dv), the solution of the homogeneous 2D Boltzmann equation without cutoff. The initial condition f0(dv) may be any probability distribution (except a Dirac mass). However, for sufﬁciently hard potentials, the semigroup has a regularization property (see Probab. Theory Related Fields 151 (2011) 659–704): ft (dv) = ft (v) dv for every t>0. The aim of this paper is to give upper bounds for ft (v), the most signiﬁcant one being of −η −|v|λ type ft (v) ≤ Ct e for some η,λ > 0.

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