## Phase transition for parking blocks, Brownian excursion and coalescence

### Cached

### Download Links

- [arxiv.org]
- [www.iecn.u-nancy.fr]
- [www.ulb.ac.be]
- [hal.archives-ouvertes.fr]
- [www.ulb.ac.be]
- [www.iecn.u-nancy.fr]
- [www.iecn.u-nancy.fr]
- [www.iecn.u-nancy.fr]
- DBLP

### Other Repositories/Bibliography

Citations: | 30 - 3 self |

### BibTeX

@MISC{Chassaing_phasetransition,

author = {P. Chassaing and G. Louchard},

title = {Phase transition for parking blocks, Brownian excursion and coalescence},

year = {}

}

### Years of Citing Articles

### OpenURL

### Abstract

In this paper, we consider hashing with linear p#neark for a hashing table with np#0W9W8 m items (m<n), and # = n -memp# yp#5j9WJ For a non comp#Rj8 science-minded reader, we shall use themetap#600 of m carsp#rskJ5 on np#0JR00 each car c i chooses ap#CW5 p i at random, and if p i is occup#WJk c i tries successively p i +1,p i + 2, until it finds anemp# yp#5RC0 Pittel [42]p#2 ves that when #/n goes to somep ositive limit #<1, the size of the largest blockof consecutive cars B n,# 1 satisfies 2(# 1 log #)B n,# 1 = 2 log 3 log log +# n , where # n converges weakly to an extreme-value distribution. In thisp#i er we examine at which level for m ap#CR8 transition occurs between B n,# 1 =o ( ) and B n,# 1 =o ( ). The intermediate case reveals an interesting behaviour of sizes of blocks, related to the standard additive coalescent in the same way as the sizes of connectedcomp onents of the randomgrap# are related to the multip#kT55WW e coalescent.

### Citations

3016 | Convergence of Probability Measures - Billingsley - 1968 |

1789 | Random Graphs
- Bollob'as
- 1981
(Show Context)
Citation Context ...gs) ; where n converges weakly to an extreme-value distribution. This paper is concerned with what we would call the "emergence of a giant block", by reference to the emergence of a giant c=-=omponent [4, 9, 14, 22, 28-=-]. We have: Theorem 1.1 For n and m going jointly to +1 (i) if p n = o(`), B n;` 1 =n P ! 0; 2 (ii) if ` = o( p n), B n;` 1 =n P ! 1. Thus a phase transition occurs for ` = ( p n). The main result of ... |

911 | Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences - Revuz, Yor - 1999 |

382 |
Empirical Processes with Applications to Statistics
- Shorack, Wellner
- 1986
(Show Context)
Citation Context ...the connection between hashing (or parking) and empirical processes. Given a sample (U 1 ; U 2 ; :::; Um ) of uniform random variables, the empirical distributionsFm and the empirical process m (see [=-=18, 4-=-3, 48] for background) are 14 respectively dened by Fm (t) = #f1 i m j U i tg m = t + m (t) p m : The process m gives a measure of the accuracy of the approximation of the true distribution functio... |

271 |
Brownian motion and stochastic flow systems
- Harrison
- 1985
(Show Context)
Citation Context ...plore connections between this limit model for parking and the reflected Brownian motion with drift, or the Brownian storage process, that appear as heavy traffic limits of queuing or storage systems =-=[27, 32]-=-. Acknowledgements The starting point for this paper was the talk of Philippe Flajolet at the meeting ALEA in February 1998 at Asnelles (concerning his paper with Viola & Poblete), and a discussion th... |

218 | The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator
- Pitman, Yor
- 1997
(Show Context)
Citation Context ...e U k are uniform on [0; 1] and independent. Actually, such a phenomenon is common for distributions of sequences related to Poisson point processes, and specially for Poisson-Dirichlet distributions =-=[10, 11, 35, 36, 4-=-1]. The distribution of R(), as described in Theorem 1.5, already appeared as the law of the -valued fragmentation process derived from the continuum random tree, introduced by Aldous & Pitman in thei... |

217 | Introduction to Analytic and Probabilistic Number Theory, Cambridge - Tenenbaum - 1995 |

142 | Deterministic and stochastic model for coalescence (aggregation and coagulation): A review of the mean-field theory and probabilists
- Aldous
- 1999
(Show Context)
Citation Context ... it can only be explained by coalescence with other blocks of size O(n), that is, by instantaneous jumps. 1.4 Coalescence We give here a brief account of coalescence, which is masterfully surveyed in =-=[5, 6-=-]. We essentially quote the two previously cited references. Models of coalescence (aggregation, coagulation, gelation ...) have been studied in many scientic disciplines, essentially physical chemist... |

124 |
Exchangeability and related topics. École d’été de probabilités de Saint-Flour
- Aldous
- 1985
(Show Context)
Citation Context ...B(), for instance it proves that almost surely each R k () is positive, and thus a.s. 0sk ()s1. Size-biased permutations of random discrete probabilities have been studied, among others, by Aldous [1]=-=-=- and Pitman [37, 38]. The most celebrated example is the size-biased permutation of the sequence of limit sizes of cycles of a random permutation. While the limit distribution of the sizes of the larg... |

107 | The continuum random tree III - Aldous - 1993 |

103 |
Exchangeable and partially exchangeable random partitions. Probab. Theory Related Fields 102(2
- Pitman
- 1995
(Show Context)
Citation Context ...nce it proves that almost surely each R k () is positive, and thus a.s. 0sk ()s1. Size-biased permutations of random discrete probabilities have been studied, among others, by Aldous [1] and Pitman [3=-=7, 38]-=-. The most celebrated example is the size-biased permutation of the sequence of limit sizes of cycles of a random permutation. While the limit distribution of the sizes of the largest, second largest ... |

84 | Brownian excursions, critical random graphs and the multiplicative coalescent
- Aldous
- 1997
(Show Context)
Citation Context ...gs) ; where n converges weakly to an extreme-value distribution. This paper is concerned with what we would call the "emergence of a giant block", by reference to the emergence of a giant c=-=omponent [4, 9, 14, 22, 28-=-]. We have: Theorem 1.1 For n and m going jointly to +1 (i) if p n = o(`), B n;` 1 =n P ! 0; 2 (ii) if ` = o( p n), B n;` 1 =n P ! 1. Thus a phase transition occurs for ` = ( p n). The main result of ... |

65 |
Size-biased sampling of Poisson point processes and excursions. Probab. Theory Related Fields 92
- Perman, Pitman, et al.
- 1992
(Show Context)
Citation Context ...e U k are uniform on [0; 1] and independent. Actually, such a phenomenon is common for distributions of sequences related to Poisson point processes, and specially for Poisson-Dirichlet distributions =-=[10, 11, 35, 36, 4-=-1]. The distribution of R(), as described in Theorem 1.5, already appeared as the law of the -valued fragmentation process derived from the continuum random tree, introduced by Aldous & Pitman in thei... |

63 | The standard additive coalescent
- Aldous, Pitman
- 1998
(Show Context)
Citation Context ... blocks' sizes is described by widths of excursions of stochastic processes related to the Brownian motion. It turns out, by nature of the problem, and also owing to previous works of Aldous & Pitman =-=[8-=-], that the description given here (specially by Theorem 1.3) is more precise than in [4]. 1.2 Phase transition and Brownian motion Recall some notations and denitions from Brownian motion theory. An ... |

57 |
The birth of the giant component,” Random Structures and Algorithms 4
- Janson, Knuth, et al.
- 1993
(Show Context)
Citation Context ...tribution. This paper is essentially concerned with what we would call the "emergence of a giant block" (see [5, 7] for an historic of the emergence of the giant component of a random graph,=-= and also [2, 14, 1-=-6]): Theorem 1.1 For n and m(n) going jointly to+1, we have: (i) if p n = o(E(n)), B (1) n =n P ! 0; (ii) if E(n) = o( p n), B (1) n =n P ! 1; (iii) if lim E(n) p n) = > 0, B (1) n =n law ! B 1 (); i... |

55 |
Ordered cycle lengths in a random permutation
- Shepp, Lloyd
- 1966
(Show Context)
Citation Context ...e-biased permutation of the sequence of limit sizes of cycles of a random permutation. While the limit distribution of the sizes of the largest, second largest ... cycle have a complicated expression =-=[19, -=-47], the successive terms R 1 ; R 2 ; ::: of their size-biased permutation satises (R 1 +R 2 + ::: +R k ) k1 law = (1 U 1 U 2 ::: U k ) k1 ; in which the U k are uniform on [0; 1] and independent. Act... |

53 |
The tight constant in the Dvoretzky-Kiefer-Wolfowitz inequality
- Massart
- 1990
(Show Context)
Citation Context ...). Actually, we have: p n n (t) = p n ` n;` (t) + p ` ~ n;` (t); with the consequence that: j n (t) n;` (t)j 1 + s 1 ` n j n;` (t)j + s ` n j ~ n;` (t)j: According to the DKW inequality [33], no=-=t-=- depending on (n; `), Pr(sup t j ~ n;` (t)j x) 2 exp( 2x 2 ); thus, for suitable K 1 and K 2 , and for " > 0, Pr sup 0` p n sup t j n (t) n;` (t)j n 1=4+" ! K 1 p n e K 2 n 2" : T... |

51 |
Brownian bridge asymptotics for random mappings
- Aldous, Pitman
- 1994
(Show Context)
Citation Context ...n Subsection 3.3, requires some care, as the sequence of widths of excursions is not a continuous functional of h n : the proof relies on an extension of the invariance principle that we learned from =-=[4, 7]-=-. Further consequences of Theorem 3.1 are Theorem 1.10 and also some results about stochastic processes developped in [16]. Theorem 1.3 is the consequence of Theorem 4.1, an extension of Theorem 3.1. ... |

48 | The continuum random tree II: an overview. Stochastic Analysis (Proc - Aldous - 1990 |

46 |
An occupancy discipline and applications
- Konheim, Weiss
- 1966
(Show Context)
Citation Context ...ces f1; 2; :::; ng, m items fc 1 ; c 2 ; :::; c m g, and ` = n m empty places (` > 0). Hashing with linear probing is a fundamental object in analysis of algorithms: its study goes back to the 1960's =-=[29, 31]-=- and is still active [2, 23, 30, 42]. For a non computer science-minded reader, we shall use, all along the paper, the metaphore of m cars parking on n places, leaving ` places empty: each car c i cho... |

45 |
On the frequency of numbers containing prime factors of a certain relative magnitude, Ark
- Dickman
- 1930
(Show Context)
Citation Context ...e-biased permutation of the sequence of limit sizes of cycles of a random permutation. While the limit distribution of the sizes of the largest, second largest ... cycle have a complicated expression =-=[19, -=-47], the successive terms R 1 ; R 2 ; ::: of their size-biased permutation satises (R 1 +R 2 + ::: +R k ) k1 law = (1 U 1 U 2 ::: U k ) k1 ; in which the U k are uniform on [0; 1] and independent. Act... |

45 | Introduction à la théorie analytique et probabiliste des nombres. Institut Elie Cartan - Tenenbaum - 1990 |

44 | Construction of Markovian coalescents
- Evans, Pitman
- 1998
(Show Context)
Citation Context ...detailing mass, position, and velocity of each cluster, is too complicated for analysis, so recent works focused on the evolution of masses of clusters through time: the general stochastic coalescent =-=-=-[21] is the continuous-time Markov process whose state space is the innite-dimensional simplex = n (x i ) i1 : x i 0; X x i = 1 o ; (the x i 's are the sizes of clusters) and that evolves according ... |

44 | Arcsine laws and interval partitions derived from a stable subordinator
- Pitman, Yor
- 1992
(Show Context)
Citation Context ...s+ ~ e that belong to the interval [g k ; d k ]: y k is also, after normalisation by x k , the decreasing sequence of widths of excursions ofs~ p x k k . As consequences of Theorem 1.10, or of [16, 40], B() and are independent, and is a sequence of independent normalized Brownian excursions. Thus, given B(), the sequences y k are independent and respectively distributed as x k B ~ p x k ... |

40 | A polytope related to empirical distributions, plane trees, parking functions, and the associahedron, Discrete Comput
- Pitman, Stanley
(Show Context)
Citation Context ...nctions and many other combinatorial objects. In this paper, we use mainly a - maybe less exploited - connection between parking functions and empirical processes of mathematical statistics (see also =-=[15, 3-=-9]) . Let B n;` k denote the size of the k th largest block of consecutive cars, and let B n;` = (B n;` k ) k1 be the decreasing sequence of sizes of blocks, ended by an innite sequence of 0's. Pittel... |

39 |
A relation between Brownian bridge and Brownian excursion
- Vervaat
- 1979
(Show Context)
Citation Context ....1 Recall that Donsker (1952), following an idea of Doob, proved that: Theorem 3.5 Let b = (b(t)) 0t1 be a Brownian bridge. We have: m weakly ! b: 16 We shall also need: Theorem 3.6 (Vervaat, 1979 [53]) Let v be the almost surely unique point such that b(v) = min 0t1 b(t). Then v is uniform and e = (e(t)) 0t1 , dened by e(t) = b(fv + tg) b(v), is a normalized Brownian excursion, independent of... |

37 | Random discrete distributions invariant under size-biased permutation
- Pitman
- 1996
(Show Context)
Citation Context ...th a probability proportional to its size, but among the terms that did not appear before. A more formal denition of size-biased permutations, by construction through a rejection method, is given in [38]: consider a sequence of independent, positive, integer-valued random variables (I k ) k1 , distributed according to B(): Pr (I k = j j B()) = B j (): With probability 1 the terms of B() are posi... |

33 | Hyperplane arrangements, parking functions and tree inversions
- Stanley
- 1996
(Show Context)
Citation Context ... 3> 2> Figure 1: An example where (n; m) = (10; 7) and B 10;3 = (3; 3; 1; 0; 0; : : : ) Under the name of parking function, hashing with linear probing has been and is still studied by combinatorists =-=[25, 26, 45, 49, 50, 51]-=-. Section 4 of [23] contains nice developments on the connections between parking functions and many other combinatorial objects. In this paper, we use mainly a - maybe less exploited - connection bet... |

31 | The birth of the giant component
- Janson, Knuth, et al.
- 1993
(Show Context)
Citation Context ...gs) ; where n converges weakly to an extreme-value distribution. This paper is concerned with what we would call the "emergence of a giant block", by reference to the emergence of a giant c=-=omponent [4, 9, 14, 22, 28-=-]. We have: Theorem 1.1 For n and m going jointly to +1 (i) if p n = o(`), B n;` 1 =n P ! 0; 2 (ii) if ` = o( p n), B n;` 1 =n P ! 1. Thus a phase transition occurs for ` = ( p n). The main result of ... |

29 |
Mappings of acyclic and parking functions
- Foata, Riordan
- 1974
(Show Context)
Citation Context ... 3> 2> Figure 1: An example where (n; m) = (10; 7) and B 10;3 = (3; 3; 1; 0; 0; : : : ) Under the name of parking function, hashing with linear probing has been and is still studied by combinatorists =-=[25, 26, 45, 49, 50, 51]-=-. Section 4 of [23] contains nice developments on the connections between parking functions and many other combinatorial objects. In this paper, we use mainly a - maybe less exploited - connection bet... |

29 |
Convergence of Stochastic Processes. Springer-Verlag. New-York
- Pollard
- 1984
(Show Context)
Citation Context ...the connection between hashing (or parking) and empirical processes. Given a sample (U 1 ; U 2 ; :::; Um ) of uniform random variables, the empirical distributionsFm and the empirical process m (see [=-=18, 4-=-3, 48] for background) are 14 respectively dened by Fm (t) = #f1 i m j U i tg m = t + m (t) p m : The process m gives a measure of the accuracy of the approximation of the true distribution functio... |

29 | Parking functions and noncrossing partitions
- Stanley
- 1997
(Show Context)
Citation Context ... 3> 2> Figure 1: An example where (n; m) = (10; 7) and B 10;3 = (3; 3; 1; 0; 0; : : : ) Under the name of parking function, hashing with linear probing has been and is still studied by combinatorists =-=[25, 26, 45, 49, 50, 51]-=-. Section 4 of [23] contains nice developments on the connections between parking functions and many other combinatorial objects. In this paper, we use mainly a - maybe less exploited - connection bet... |

28 |
A fragmentation process connected to Brownian motion
- BERTOIN
- 2000
(Show Context)
Citation Context ...cess, has the same law as the process of hitting times of the Brownian motion. Incidentally, let L() denote the length of the excursion ofs e beginning at 0, and set ~ () = 1 + 1 L() : Bertoin [12] nicely proves that ~ has the same law as the process of hitting times of the Brownian motion. For the moment, we do not see any combinatorial explanation of this identity between R and L. As we tal... |

25 |
Poisson process approximations for the Ewens sampling formula
- Arratia, Barbour, et al.
- 1992
(Show Context)
Citation Context ...e U k are uniform on [0; 1] and independent. Actually, such a phenomenon is common for distributions of sequences related to Poisson point processes, and specially for Poisson-Dirichlet distributions =-=[10, 11, 35, 36, 4-=-1]. The distribution of R(), as described in Theorem 1.5, already appeared as the law of the -valued fragmentation process derived from the continuum random tree, introduced by Aldous & Pitman in thei... |

25 | Justification and extension of Doob’s heuristic approach to the KomogorovSmirnov theorems - Donsker - 1952 |

20 | Parking functions, empirical processes, and the width of rooted labeled trees, Preprint available via http://www.iecn.u-nancy.fr/∼chassain//theme.html
- Chassaing
- 1999
(Show Context)
Citation Context ...nctions and many other combinatorial objects. In this paper, we use mainly a - maybe less exploited - connection between parking functions and empirical processes of mathematical statistics (see also =-=[15, 3-=-9]) . Let B n;` k denote the size of the k th largest block of consecutive cars, and let B n;` = (B n;` k ) k1 be the decreasing sequence of sizes of blocks, ended by an innite sequence of 0's. Pittel... |

17 | Linear Probing and Graphs
- Knuth
- 1998
(Show Context)
Citation Context ...c 1 ; c 2 ; :::; c m g, and ` = n m empty places (` > 0). Hashing with linear probing is a fundamental object in analysis of algorithms: its study goes back to the 1960's [29, 31] and is still active =-=[2, 23, 30, 42]-=-. For a non computer science-minded reader, we shall use, all along the paper, the metaphore of m cars parking on n places, leaving ` places empty: each car c i chooses a place p i at random, and if p... |

17 | Local Times and Excursions for Brownian Motion: A Concise Introduction - Yor - 1995 |

16 |
The asymptotic distribution of maximum tree size in a random
- Pavlov
- 1977
(Show Context)
Citation Context ... 2k expf 4 =2( 2 x 1 : : : x k )gdx 1 : : : dx k (2) k=2 (x 1 : : : x k ( 2 x 1 : : : x k )) 3=2 ; in which D(; x; k) = n (x i ) 1ik : x i 2 x; 1 i k; and X x i 2 o : Theorem 4 of [34] gives the limit law of the largest tree in a random forest: it turns out that forests and parking schemes are in one-to-one correspondence (see Subsection 5.1). Flajolet & Salvy [24] have a direct ap... |

12 |
Acyclic and parking functions
- Françon
- 1975
(Show Context)
Citation Context |

10 | The probabilistic method. With an appendix by Paul Erdos - Alon, Spencer - 1992 |

10 |
On an enumeration problem
- Schutzenberger
- 1968
(Show Context)
Citation Context |

10 | A Vervaat-like path transformation for the reflected Brownian bridge conditioned on its local time at 0
- Chassaing, Janson
(Show Context)
Citation Context ...ame distribution as τwΨ λ √ 1−x Actually, not only the conditional distribution of τwr, but also the conditional distribution of r has a simple description in terms of the Brownian motion, and also e =-=[16]-=-. However, the weaker form (iii) fills as a nonuniform random shift of Ψ λ √ 1−x our needs for the proofs of Theorems 1.5, 1.8 and 1.9. The paper is organized as follows. Section 2 analyses the block ... |

9 |
On the Analysis of Linear Probing
- Flajolet, Poblete, et al.
- 1998
(Show Context)
Citation Context ...c 1 ; c 2 ; :::; c m g, and ` = n m empty places (` > 0). Hashing with linear probing is a fundamental object in analysis of algorithms: its study goes back to the 1960's [29, 31] and is still active =-=[2, 23, 30, 42]-=-. For a non computer science-minded reader, we shall use, all along the paper, the metaphore of m cars parking on n places, leaving ` places empty: each car c i chooses a place p i at random, and if p... |

8 |
Order statistics for jumps of normalised subordinators
- Perman
- 1993
(Show Context)
Citation Context |

8 | Introduction àlathéorie analytique et probabiliste des nombres, Société Mathématique de - Tenenbaum - 1990 |

7 |
Strong approximations in probability and statistics. Probability and Mathematical Statistics
- Csörgő, Révész
- 1981
(Show Context)
Citation Context ...the connection between hashing (or parking) and empirical processes. Given a sample (U 1 ; U 2 ; :::; Um ) of uniform random variables, the empirical distributionsFm and the empirical process m (see [=-=18, 4-=-3, 48] for background) are 14 respectively dened by Fm (t) = #f1 i m j U i tg m = t + m (t) p m : The process m gives a measure of the accuracy of the approximation of the true distribution functio... |

7 |
Knuth,The art of computer programming. Vol. 3: sorting and searching
- E
- 1998
(Show Context)
Citation Context ...ces f1; 2; :::; ng, m items fc 1 ; c 2 ; :::; c m g, and ` = n m empty places (` > 0). Hashing with linear probing is a fundamental object in analysis of algorithms: its study goes back to the 1960's =-=[29, 31]-=- and is still active [2, 23, 30, 42]. For a non computer science-minded reader, we shall use, all along the paper, the metaphore of m cars parking on n places, leaving ` places empty: each car c i cho... |

7 |
probing: the probable largest search time grows logarithmically with the number of records
- Pittel, Linear
- 1987
(Show Context)
Citation Context ...shall use the metaphore of m cars parking on n places: each car c i chooses a place p i at random, and if p i is occupied, c i tries successively p i + 1, p i + 2, until itsnds an empty place. Pittel =-=[-=-42] proves that when `=n goes to some positive limits1, the size of the largest block of consecutive cars B n;` 1 satises 2(s1 logs)B n;` 1 = 2 log n 3 log log n + n , where n converges weakly to an e... |

6 |
On Poisson-Dirichlet limits for random decomposable combinatorial structures
- Arratia, Barbour, et al.
- 1999
(Show Context)
Citation Context |