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Condensation and Extremes for a Fluctuating Number of Independent Random Variables

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Abstract

We address the question of condensation and extremes for three classes of intimately related stochastic processes: (a) random allocation models and zero-range processes, (b) tied-down renewal processes, (c) free renewal processes. While for the former class the number of components of the system is fixed, for the two other classes it is a fluctuating quantity. Studies of these topics are scattered in the literature and usually dressed up in other clothing. We give a stripped-down account of the subject in the language of sums of independent random variables in order to free ourselves of the consideration of particular models and highlight the essentials. Besides giving a unified presentation of the theory, this work investigates facets so far unexplored in previous studies. Specifically, we show how the study of the class of random allocation models and zero-range processes can serve as a backdrop for the study of the two other classes of processes central to the present work—tied-down and free renewal processes. We then present new insights on the extreme value statistics of these three classes of processes which allow a deeper understanding of the mechanism of condensation and the quantitative analysis of the fluctuations of the condensate.

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Notes

  1. In the further course of this work, the symbol \(\approx \) stands for asymptotic equivalence; the symbol \(\sim \) means either ‘of the order of’, or ‘with exponential accuracy’, depending on the context.

  2. The fact that condensation also occurs when \({\theta }<1\) in the present context has been previously mentioned in [45]. I am indebted to S Grosskinsky for pointing this reference to me.

  3. For the sake of simplicity we restrict the study to the case where f(k) is a normalisable probability distribution.

  4. Equation (7.24) corrects the inaccurate expression (4.74) given in [28] for this quantity.

  5. Whenever no ambiguity arises, we use the same notations for the observables of the tied-down and free renewal processes. Otherwise, when necessary, we add a superscript, as e.g., for \(Z^{{\scriptstyle \mathrm {td}}}\), \(Z^{{\scriptstyle \mathrm {f}}}\) or in (9.18).

  6. After submission of the present work, a study devoted to the statistics of \(X_\mathrm{max}\) in the range (L/2, L) for tied-down or free renewal processes at criticality (\(w=1\)) was presented in [55]. For the tdrp case the result (4.48) with \(w=1\) is obtained. For the free renewal case, [55] predicts, if \(w=1\),

    figure a

    which is (9.57), with \(w=1\), noting that \(Z^{{\scriptstyle \mathrm {td}}}(1,L)=\langle N^{{\scriptstyle \mathrm {f}}}_L\rangle -\langle N^{{\scriptstyle \mathrm {f}}}_{L-1}\rangle \), as is clear by taking the generating functions of both sides.

  7. I am indebted to M Loulakis for sharing his comments on this part with me.

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Acknowledgements

It is a pleasure to thank G Giacomin, M Loulakis and J-M Luck for enlightening discussions. I am also indebted to S Grosskinsky and S Janson for useful correspondence.

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Correspondence to Claude Godrèche.

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Appendices

Appendix

A On Equation (3.18)

Let us explain the argument leading to (3.18) and the origin of the hierarchical structure mentioned in Sect. 3.3.Footnote 7

  1. 1.

    In the uphill region where \(L-X_\mathrm{max}\) is finite, \(X_\mathrm{max}\) is the unique big summand and (3.16) holds. This property stems from the fact that when the sum of n subexponential random variables is conditioned to a large value L, all the dependence is absorbed by the maximum and the ensemble of \(n-1\) smaller variables becomes asymptotically independent. This property, initially put forward by early workers, has been progressively refined in subsequent studies [3, 16, 19].

  2. 2.

    In the dip region, where \(X_\mathrm{max}>L/2\), since \(L-X_\mathrm{max}\) gets large, the sum \(\sum _{i=1}^{n-1}X_i\) becomes subjected to a large deviation event. This event will be realised by \(X^{(2)}\), the second largest summand, typically equal to \(L-X_\mathrm{max}\). We thus obtain (3.17).

  3. 3.

    One can now iterate the reasoning. If \(X_\mathrm{max}=k\le L/2\), the difference \(L-k\ge k\) cannot accommodate a single big summand \(X^{(2)}=j\) since the latter should be less than \(X_\mathrm{max}\). Now

    $$\begin{aligned} L-X_\mathrm{max}-X^{(2)}\mathrel {\mathop {\approx }\limits _{L\rightarrow \infty }}\sum _{i=1}^{n-2}X_i. \end{aligned}$$
    (A.1)

    where the sum in the right side is subjected to a large deviation, which will be realised by a third large summand \(X^{(3)}\). Since \(L-k-j\) should be less than j, the constraint \(j\ge (L-k)/2\) holds. Moreover \(X^{(2)}\le X_\mathrm{max}\) imposes the condition \((L-k)/2\le k\), i.e. \(k\ge L/3\). We are thus lead to the asymptotic estimate

    $$\begin{aligned} \mathrm{Prob}(X_\mathrm{max}=k|S_n=L)\approx \sum _{j=\frac{L-k}{2}}^{k}n(n-1)\frac{f(k)f(j)Z_{n-2}(L-k-j)}{Z_n(L)}, \end{aligned}$$
    (A.2)

    and therefore

    $$\begin{aligned} \mathrm{Prob}(L/3\le X_\mathrm{max}\le L/2|S_n=L)\mathrel {\mathop {\approx }\limits _{}^{}} \sum _{k=L/3}^{L/2}\ \sum _{j=\frac{L-k}{2}}^k n(n-1)\frac{f(k)f(j)Z_{n-2}(L-k-j)}{Z_n(L)}. \end{aligned}$$
    (A.3)

For \({\theta }<1\), the analysis of this expression in the continuum limit leads to

$$\begin{aligned} \mathrm{Prob}(L/3\le X_\mathrm{max}\le L/2|S_n=L)\approx (n-1)(n-2)c^2\frac{A({\theta })}{L^{2{\theta }}}, \end{aligned}$$
(A.4)

where the amplitude \(A({\theta })\) is given by

$$\begin{aligned} A({\theta })=\int _{1/3}^{1/2}\mathrm{d }x\, \int _{\frac{1-x}{2}}^{x}\mathrm{d }y\,\frac{1}{[xy(1-x-y)]^{1+{\theta }}}. \end{aligned}$$
(A.5)

For instance, \(A(1/2)=2\pi \), \(A(1/3)=9\sqrt{3}\,\varGamma (2/3)^3/(4\pi )\).

For \({\theta }>1\), the analysis of (A.3) yields

$$\begin{aligned} \mathrm{Prob}(L/3<X_\mathrm{max}<L/2|S_n=L)\approx \frac{(n-1)c\,2^{2+2{\theta }}}{L^{1+{\theta }}} \frac{(n-2)\langle X\rangle }{2}, \end{aligned}$$
(A.6)

for continuous random variables, and

$$\begin{aligned} \mathrm{Prob}(L/3\le X_\mathrm{max}\le L/2|S_n=L)\approx \frac{(n-1)c\,2^{2+2{\theta }}}{L^{1+{\theta }}} \frac{(n-2)\langle X\rangle +1}{2}, \end{aligned}$$
(A.7)

for discrete ones.

One can iterate the reasoning leading to (A.3) and derive the weights of the successive sectors (L/4, L/3), (L/5, L/4), etc. For instance, for \(X_\mathrm{max}\) in the interval (L/4, L/3), one finds

$$\begin{aligned} \mathrm{Prob}(X_\mathrm{max}=k|S_n=L)\approx \sum _{j=\frac{L-k}{3}}^{k}\sum _{i=\frac{L-k-j}{2}}^jn(n-1)(n-2)\frac{f(k)f(j)f(i)Z_{n-3}(L-k-j-i)}{Z_n(L)}, \end{aligned}$$
(A.8)

which yields

$$\begin{aligned} \mathrm{Prob}(L/4\le X_\mathrm{max}\le L/3|S_n=L)\sim L^{-\gamma },\qquad \gamma = \left\{ \begin{array}{ll} 3{\theta }\ &{} \text {if } {\theta }\le 2\\ 2(1+{\theta })\ &{} \text {if }{\theta }>2, \end{array} \right. \end{aligned}$$
(A.9)

For example, if \({\theta }=1/2\), one finds

$$\begin{aligned} \mathrm{Prob}(L/4\le X_\mathrm{max}\le L/3|S_n=L)\approx (n-1)(n-2)(n-3)c^3\frac{\sqrt{2}\pi /3}{L^{3/2}}. \end{aligned}$$
(A.10)

B Weight of the Maximum in the Left Region for a Lévy \(\frac{1}{2}\) Stable Law

We want to determine the weight of the maximum in the left region considered in Sect. 3.3,

$$\begin{aligned} P_n= & {} \mathrm{Prob}(X_\mathrm{max}\le L/2|S_n=L)=\sum _{k=0}^{L/2}p_n(k|L)=1-\sum _{k=L/2+1}^{L}p_n(k|L)\nonumber \\= & {} 1-\sum _{k=L/2+1}^{L}n\pi _n(k|L), \end{aligned}$$
(B.1)

on the particular example of a a Lévy \(\frac{1}{2}\) stable law. We use a continuum formalism where distributions are densities and variables are real numbers for the particular case where f(k) is the Lévy \(\frac{1}{2}\) stable density (7.11),

$$\begin{aligned} f(k)=\mathcal {L}_{\frac{1}{2},c}(k)=\frac{c}{k^{3/2}}\mathrm{e}^{-\pi c^2/k}. \end{aligned}$$
(B.2)

Likewise, considering L as a real number and \(Z_n(L)\) as a density,

$$\begin{aligned} Z_n(L)=\mathcal {L}_{\frac{1}{2},nc}(k)=\frac{nc}{L^{3/2}}\mathrm{e}^{-n^2\pi c^2/L}. \end{aligned}$$
(B.3)

Thus

$$\begin{aligned} \pi _n(k|L)=\frac{f(k)Z_{n-1}(L-k)}{Z_n(L)} \end{aligned}$$
(B.4)

is explicit. Setting \(k=L/t\), we obtain

$$\begin{aligned} P_n=1-\frac{(n-1)c}{\sqrt{L}}\int _{1}^2\mathrm{d }t\,\frac{t\,\mathrm{e}^{-\frac{\pi c^2(n-t)^2}{L(t-1)}}}{(t-1)^{3/2}}. \end{aligned}$$
(B.5)

Setting \(c/\sqrt{L}=b/(2\sqrt{\pi })\), we finally get

$$\begin{aligned} P_n=\frac{1}{2}\Bigg (2-n\mathrm{erfc}\Big (\frac{(n-2)b}{2}\Big )+(n-2)\mathrm{e}^{(n-1)b^2}\mathrm{erfc}\Big (\frac{nb}{2}\Big ) \Bigg ). \end{aligned}$$
(B.6)

For L large, expanding in powers of b, we obtain

$$\begin{aligned} P_n= & {} (n-1)(n-2)\Big (\frac{b^2}{2}-n\frac{b^3}{3\sqrt{\pi }}+(n-1)\frac{b^4}{4}+\cdots \Big ) \nonumber \\= & {} (n-1)(n-2)\Big (2\pi \frac{c^2}{L}-\frac{8\pi n}{3}\frac{c^3}{L^{3/2}}+(n-1)4\pi ^2\frac{c^4}{L^2}+\cdots \Big ). \end{aligned}$$
(B.7)

For Example 1, \(c=1/(2\sqrt{\pi })\) (see (5.1)), the first term of the expansion,

$$\begin{aligned} P_n\approx \frac{(n-1)(n-2)}{2L}, \end{aligned}$$
(B.8)

matches the predictions made in (A.4) and (A.5).

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Godrèche, C. Condensation and Extremes for a Fluctuating Number of Independent Random Variables. J Stat Phys 182, 13 (2021). https://doi.org/10.1007/s10955-020-02679-w

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