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Janson, Svante

Open this publication in new window or tab >>Patterns in Random Permutations Avoiding Some Sets of Multiple Patterns### Janson, Svante

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##### Abstract [en]

##### Place, publisher, year, edition, pages

SPRINGER, 2020
##### Keywords

Random permutations, Patterns in permutations, Forbidden patterns
##### National Category

Computer Sciences Discrete Mathematics
##### Identifiers

urn:nbn:se:uu:diva-406496 (URN)10.1007/s00453-019-00586-5 (DOI)000511594700006 ()
#####

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##### Funder

Knut and Alice Wallenberg Foundation
Available from: 2020-03-10 Created: 2020-03-10 Last updated: 2020-03-10Bibliographically approved

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.

We consider a random permutation drawn from the set of permutations of length n that avoid some given set of patterns of length 3. We show that the number of occurrences of another pattern sigma has a limit distribution, after suitable scaling. In several cases, the number is asymptotically normal; this contrasts to the cases of permutations avoiding a single pattern of length 3 studied in earlier papers.

Open this publication in new window or tab >>A modified bootstrap percolation on a random graph coupled with a lattice### Janson, Svante

### Kozma, Robert

### Ruszinko, Miklos

### Sokolov, Yury

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_some",{id:"formSmash:j_idt184:1:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_otherAuthors",{id:"formSmash:j_idt184:1:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_otherAuthors",multiple:true}); 2019 (English)In: Discrete Applied Mathematics, ISSN 0166-218X, E-ISSN 1872-6771, Vol. 258, p. 152-165Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

Graph diameter, Degree distribution, Bootstrap percolation, Phase transitions
##### National Category

Probability Theory and Statistics Discrete Mathematics
##### Identifiers

urn:nbn:se:uu:diva-382249 (URN)10.1016/j.dam.2018.11.006 (DOI)000462694000015 ()
#####

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##### Funder

Knut and Alice Wallenberg Foundation, 2016.0357
Available from: 2019-05-03 Created: 2019-05-03 Last updated: 2019-05-03Bibliographically approved

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.

Univ Memphis, Dept Math, Memphis, TN 38152 USA;Univ Massachusetts, Dept Comp Sci, Amherst, MA 01003 USA.

Hungarian Acad Sci, Alfed Renyi Inst Math, 13-15 Realtanoda Utca, H-1053 Budapest, Hungary.

Univ Calif San Diego, Dept Med, 9300 Campus Point,Dr 7374, La Jolla, CA 92037 USA.

In this paper a random graph model G(ZN2.pd) is introduced, which is a combination of fixed torus grid edges in (z/NZ)(2) and some additional random ones. The random edges are called long, and the probability of having a long edge between vertices u, v is an element of (Z/NZ)(2) with graph distance d on the torus grid is p(d) = c/Nd, where c is some constant. We show that, whp, the diameter D(G(zN2.pd)) = Theta(log N). Moreover, we consider a modified non-monotonous bootstrap percolation on G(ZN2.pd). We prove the presence of phase transitions in mean-field approximation and provide Fairly sharp bounds on the error of the critical parameters.

Open this publication in new window or tab >>A piecewise contractive dynamical system and Phragmèn's election method### Janson, Svante

### Öberg, Anders

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_some",{id:"formSmash:j_idt184:2:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_otherAuthors",{id:"formSmash:j_idt184:2:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_otherAuthors",multiple:true}); 2019 (English)In: Bulletin de la Société Mathématique de France, ISSN 0037-9484, E-ISSN 2102-622X, Vol. 147, no 3, p. 395-441Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

FRENCH MATHEMATICAL SOC, 2019
##### Keywords

piecewise contractive dynamical system, Phragmens election method
##### National Category

Mathematical Analysis
##### Identifiers

urn:nbn:se:uu:diva-396947 (URN)10.24033/bsmf.2787 (DOI)000492745100003 ()
#####

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Available from: 2019-11-21 Created: 2019-11-21 Last updated: 2019-11-21Bibliographically approved

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.

We prove some basic results for a dynamical system given by a piece-wise linear and contractive map on the unit interval that takes two possible values at a point of discontinuity. We prove that there exists a universal limit cycle in the non-exceptional cases, and that the exceptional parameter set is very tiny in terms of gauge functions. The exceptional two-dimensional parameter is shown to have Hausdorff-dimension one. We also study the invariant sets and the limit sets; these are sometimes different and there are several cases to consider. In addition, we prove the existence of a unique invariant measure. We apply some of our results for the dynamical system, involving a study of rational and irrational rotation numbers, to a combinatorial problem involving an election method suggested by Phragmen, and we show that the proportion of elected seats for each party converges to a limit, which is a rational number except for a very small exceptional set of parameters.

Open this publication in new window or tab >>Competing first passage percolation on random graphs with finite variance degrees### Ahlberg, Daniel

### Deijfen, Maria

### Janson, Svante

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_some",{id:"formSmash:j_idt184:3:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_otherAuthors",{id:"formSmash:j_idt184:3:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_otherAuthors",multiple:true}); 2019 (English)In: Random structures & algorithms (Print), ISSN 1042-9832, E-ISSN 1098-2418, Vol. 55, no 3, p. 545-559Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

WILEY, 2019
##### Keywords

coexistence, competing growth, configuration model, continuous-time branching process, first passage percolation, random graphs
##### National Category

Computer Sciences Probability Theory and Statistics
##### Identifiers

urn:nbn:se:uu:diva-393719 (URN)10.1002/rsa.20846 (DOI)000482128300002 ()
#####

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Available from: 2019-09-26 Created: 2019-09-26 Last updated: 2019-09-26Bibliographically approved

Stockholm Univ, Dept Math, S-10691 Stockholm, Sweden.

Stockholm Univ, Dept Math, S-10691 Stockholm, Sweden.

We study the growth of two competing infection types on graphs generated by the configuration model with a given degree sequence. Starting from two vertices chosen uniformly at random, the infection types spread via the edges in the graph in that an uninfected vertex becomes type 1 (2) infected at rate lambda(1) (lambda(2)) times the number of nearest neighbors of type 1 (2). Assuming (essentially) that the degree of a randomly chosen vertex has finite second moment, we show that if lambda(1) = lambda(2), then the fraction of vertices that are ultimately infected by type 1 converges to a continuous random variable V is an element of (0,1), as the number of vertices tends to infinity. Both infection types hence occupy a positive (random) fraction of the vertices. If lambda(1) not equal lambda(2), on the other hand, then the type with the larger intensity occupies all but a vanishing fraction of the vertices. Our results apply also to a uniformly chosen simple graph with the given degree sequence.

Open this publication in new window or tab >>Competition in growth and urns### Ahlberg, Daniel

### Griffiths, Simon

### Janson, Svante

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.### Morris, Robert

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_4_j_idt188_some",{id:"formSmash:j_idt184:4:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_4_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_4_j_idt188_otherAuthors",{id:"formSmash:j_idt184:4:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_4_j_idt188_otherAuthors",multiple:true}); 2019 (English)In: Random structures & algorithms (Print), ISSN 1042-9832, E-ISSN 1098-2418, Vol. 54, no 2, p. 211-227Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

WILEY, 2019
##### Keywords

bootstrap percolation, branching processes, competing growth, urn models
##### National Category

Probability Theory and Statistics
##### Identifiers

urn:nbn:se:uu:diva-378525 (URN)10.1002/rsa.20779 (DOI)000458197400001 ()
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##### Funder

Swedish Research Council, 637-2013-7302Knut and Alice Wallenberg FoundationEU, European Research Council, 680275 MALIG
Available from: 2019-03-25 Created: 2019-03-25 Last updated: 2019-03-25Bibliographically approved

Stockholm Univ, Dept Math, SE-10691 Stockholm, Sweden.

PUC Rio, Dept Matemat, BR-22451900 Gavea, RJ, Brazil.

Inst Nacl Matemat Pura & Aplicada, Estr Dona Castorina 110, BR-22460320 Rio De Janeiro, Brazil.

We study survival among two competing types in two settings: a planar growth model related to two-neighbor bootstrap percolation, and a system of urns with graph-based interactions. In the planar growth model, uncolored sites are given a color at rate 0, 1 or infinity, depending on whether they have zero, one, or at least two neighbors of that color. In the urn scheme, each vertex of a graph G has an associated urn containing some number of either blue or red balls ( but not both). At each time step, a ball is chosen uniformly at random from all those currently present in the system, a ball of the same color is added to each neighboring urn, and balls in the same urn but of different colors annihilate on a one-for-one basis. We show that, for every connected graph G and every initial configuration, only one color survives almost surely. As a corollary, we deduce that in the two-type growth model on Z(2), one of the colors only infects a finite number of sites with probability one. We also discuss generalizations to higher dimensions and multi-type processes, and list a number of open problems and conjectures.

Open this publication in new window or tab >>Component structure of the configuration model: Barely supercritical case### van der Hofstad, Remco

### Janson, Svante

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.### Luczak, Malwina

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##### Abstract [en]

##### Place, publisher, year, edition, pages

WILEY, 2019
##### Keywords

percolation, phase transition, random graphs, scaling window
##### National Category

Computer Sciences
##### Identifiers

urn:nbn:se:uu:diva-390078 (URN)10.1002/rsa.20837 (DOI)000471321600001 ()
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##### Funder

Knut and Alice Wallenberg Foundation
Available from: 2019-08-06 Created: 2019-08-06 Last updated: 2019-08-06Bibliographically approved

Eindhoven Univ Technol, Dept Math & Comp Sci, Eindhoven, Netherlands.

Univ Melbourne, Sch Math & Stat, Parkville, Vic, Australia.

We study near-critical behavior in the configuration model. Let D-n be the degree of a random vertex and nu n=E[Dn(Dn-1)]/E[Dn]; we consider the barely supercritical regime, where nu(n)-> 1 as n ->infinity, but nu n-1 n-1/3(E[Dn3])2/3. Let Dn* denote the size-biased version of D-n. We prove that there is a unique giant component of size n rho nEDn(1+o(1)), where rho(n) denotes the survival probability of a branching process with offspring distribution Dn*-1. This extends earlier results of Janson and Luczak, as well as those of Janson, Luczak, Windridge, and House, to the case where the third moment of D-n is unbounded. We further study the size of the largest component in the critical regime, where nu n-1=O(n-1/3(EDn3)2/3), extending and complementing results of Hatami and Molloy.

Open this publication in new window or tab >>Inversions in Split Trees and Conditional Galton-Watson Treest### Cai, Xing Shi

### Holmgren, Cecilia

### Janson, Svante

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.### Johansson, Tony

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_6_j_idt188_some",{id:"formSmash:j_idt184:6:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_6_j_idt188_some",multiple:true}); ### Skerman, Fiona

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_6_j_idt188_otherAuthors",{id:"formSmash:j_idt184:6:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_6_j_idt188_otherAuthors",multiple:true}); Show others...PrimeFaces.cw("SelectBooleanButton","widget_formSmash_j_idt184_6_j_idt188_j_idt202",{id:"formSmash:j_idt184:6:j_idt188:j_idt202",widgetVar:"widget_formSmash_j_idt184_6_j_idt188_j_idt202",onLabel:"Hide others...",offLabel:"Show others..."}); 2019 (English)In: Combinatorics, probability & computing, ISSN 0963-5483, E-ISSN 1469-2163, Vol. 28, no 3, p. 335-364Article in journal (Refereed) Published
##### Abstract [en]

##### National Category

Probability Theory and Statistics Computer Sciences
##### Identifiers

urn:nbn:se:uu:diva-382243 (URN)10.1017/S0963548318000512 (DOI)000462848800002 ()
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##### Funder

Knut and Alice Wallenberg FoundationSwedish Research Council
Available from: 2019-05-15 Created: 2019-05-15 Last updated: 2019-05-15Bibliographically approved

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.

We study I(T), the number of inversions in a tree T with its vertices labelled uniformly at random, which is a generalization of inversions in permutations. We first show that the cumulants of I(T) have explicit formulas involving the k-total common ancestors of T (an extension of the total path length). Then we consider X-n, the normalized version of I(T-n), for a sequence of trees T-n. For fixed T-n's, we prove a sufficient condition for X-n to converge in distribution. As an application, we identify the limit of X-n for complete b-ary trees. For T(n )being split trees [16], we show that X-n converges to the unique solution of a distributional equation. Finally, when T-n's are conditional Galton-Watson trees, we show that X-n converges to a random variable defined in terms of Brownian excursions. By exploiting the connection between inversions and the total path length, we are able to give results that significantly strengthen and broaden previous work by Panholzer and Seitz [46].

Open this publication in new window or tab >>Long Term Behaviour of a Reversible System of Interacting Random Walks### Janson, Svante

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.### Shcherbakov, Vadim

### Volkov, Stanislav

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_7_j_idt188_some",{id:"formSmash:j_idt184:7:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_7_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_7_j_idt188_otherAuthors",{id:"formSmash:j_idt184:7:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_7_j_idt188_otherAuthors",multiple:true}); 2019 (English)In: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 175, no 1, p. 71-96Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

SPRINGER, 2019
##### Keywords

Markov chain, Random walk, Transience, Recurrence, Lyapunov function, Martingale, Renewal measure, Return time
##### National Category

Probability Theory and Statistics
##### Identifiers

urn:nbn:se:uu:diva-381570 (URN)10.1007/s10955-019-02244-0 (DOI)000462211500004 ()
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##### Funder

Knut and Alice Wallenberg FoundationSwedish Research Council, VR2014-5157
Available from: 2019-04-12 Created: 2019-04-12 Last updated: 2019-04-12Bibliographically approved

Royal Holloway Univ London, Egham, Surrey, England.

Lund Univ, Lund, Sweden.

This paper studies the long-term behaviour of a system of interacting random walks labelled by vertices of a finite graph. We show that the system undergoes phase transitions, with different behaviour in various regions, depending on model parameters and properties of the underlying graph. We provide the complete classification of the long-term behaviour of the corresponding continuous time Markov chain, identifying whether it is null recurrent, positive recurrent, or transient. The proofs are partially based on the reversibility of the model, which allows us to use the method of electric networks. We also provide some alternative proofs (based on the Lyapunov function method and the renewal theory), which are of interest in their own right, since they do not require reversibility and can be applied to more general situations.

Open this publication in new window or tab >>Patterns in random permutations avoiding the pattern 321### Janson, Svante

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_8_j_idt188_some",{id:"formSmash:j_idt184:8:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_8_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_8_j_idt188_otherAuthors",{id:"formSmash:j_idt184:8:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_8_j_idt188_otherAuthors",multiple:true}); 2019 (English)In: Random structures & algorithms (Print), ISSN 1042-9832, E-ISSN 1098-2418, Vol. 55, no 2, p. 249-270Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

WILEY, 2019
##### Keywords

patterns in permutations, pattern-avoiding permutations, random permutations
##### National Category

Computer Sciences
##### Identifiers

urn:nbn:se:uu:diva-390378 (URN)10.1002/rsa.20806 (DOI)000475814600001 ()
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PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_8_j_idt188_j_idt359",{id:"formSmash:j_idt184:8:j_idt188:j_idt359",widgetVar:"widget_formSmash_j_idt184_8_j_idt188_j_idt359",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_8_j_idt188_j_idt371",{id:"formSmash:j_idt184:8:j_idt188:j_idt371",widgetVar:"widget_formSmash_j_idt184_8_j_idt188_j_idt371",multiple:true});
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##### Funder

Knut and Alice Wallenberg Foundation
Available from: 2019-08-12 Created: 2019-08-12 Last updated: 2019-08-12Bibliographically approved

We consider a random permutation drawn from the set of 321-avoiding permutations of length n and show that the number of occurrences of another pattern sigma has a limit distribution, after scaling by n(m + l) where m is the length of sigma and l is the number of blocks in it. The limit is not normal, and can be expressed as a functional of a Brownian excursion.

Open this publication in new window or tab >>Preferential attachment when stable### Janson, Svante

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.### Sen, Subhabrata

### Spencer, Joel

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_9_j_idt188_some",{id:"formSmash:j_idt184:9:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_9_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_9_j_idt188_otherAuthors",{id:"formSmash:j_idt184:9:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_9_j_idt188_otherAuthors",multiple:true}); 2019 (English)In: Advances in Applied Probability, ISSN 0001-8678, E-ISSN 1475-6064, Vol. 51, no 4, p. 1067-1108Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

APPLIED PROBABILITY TRUST, 2019
##### Keywords

Urn model, large deviations
##### National Category

Probability Theory and Statistics
##### Identifiers

urn:nbn:se:uu:diva-398564 (URN)10.1017/apr.2019.42 (DOI)000496968500005 ()
#####

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PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_9_j_idt188_j_idt371",{id:"formSmash:j_idt184:9:j_idt188:j_idt371",widgetVar:"widget_formSmash_j_idt184_9_j_idt188_j_idt371",multiple:true});
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##### Funder

Knut and Alice Wallenberg Foundation
Available from: 2019-12-09 Created: 2019-12-09 Last updated: 2019-12-09Bibliographically approved

Harvard Univ, Dept Stat, 1 Oxford St,SC 712, Cambridge, MA 02138 USA.

NYU, Courant Inst, Dept Comp Sci, Room 829,251 Mercer St, New York, NY 10012 USA;NYU, Courant Inst, Dept Math, Room 829,251 Mercer St, New York, NY 10012 USA.

We study an urn process with two urns, initialized with a ball each. Balls are added sequentially, the urn being chosen independently with probability proportional to the ath power (alpha > 1) of the existing number of balls. We study the (rare) event that the urn compositions are balanced after the addition of 2n - 2 new balls. We derive precise asymptotics of the probability of this event by embedding the process in continuous time. Quite surprisingly, fine control of this probability may be leveraged to derive a lower- tail large deviation principle (LDP) for L= Sigma(n)(i=1) (S-i(2)/i(2)), where {S-n : n >= 0} is a simple symmetric random walk started at zero. We provide an alternative proof of the LDP via coupling to Brownian motion, and subsequent derivation of the LDP for a continuous-time analog of L. Finally, we turn our attention back to the urn process conditioned to be balanced, and provide a functional limit law describing the trajectory of the urn process.