Fluctuations in the random-link matching problem

TL;DR

This paper analyzes the fluctuations of the optimal cost in the random-link matching problem using advanced statistical physics methods, deriving exact variance and large deviation functions.

Contribution

It provides the first exact expressions for the variance and large deviation functions in the random-link matching problem.

Findings

Exact variance of optimal cost derived

Large deviation function characterized

Methods applicable to related combinatorial problems

Abstract

Using the replica approach and the cavity method, we study the fluctuations of the optimal cost in the random-link matching problem. By means of replica arguments, we derive the exact expression of its variance. Moreover, we study the large deviation function, deriving its expression in two different ways, namely using both the replica method and the cavity method.

Figures

Equations

$$C_N[\mathbf{M}]:=\sum _{i<j}{m}_{ij}{w}_{ij}$$

$$m_{ji}=m_{ij}\in \{0,1\}\forall i,j,\sum _{i=1}^{2N}{m}_{ij}=1\forall j.$$

$$\mathcal{C}:=\underset{N\to +\infty }{\lim }\mathbb{E}\left[\underset{\mathbf{M}}{\min }{C}_N[\mathbf{M}]\right]=\frac{\zeta (2)}{2},$$

$$Z_w(\beta ):=\sum _{\{m_{ij}\}}\prod _{i=1}^{2N}\mathbb{I}\left(\sum _{j=1}^{2N}{m}_{ij}=1\right){\mathrm{e}}^{-\beta N{C}_N[\mathbf{M}]}$$

$$\Phi (n,\beta ):=-\underset{N\to +\infty }{\lim }\frac{\ln \mathbb{E}\left[Z_w^n(\beta )\right]}{\beta nN}.$$

$$\begin{gathered} -\beta n\Phi (n,\beta )=-\frac{\beta }{2}\sum _{p=1}^n\frac{\Gamma (n+1)}{\Gamma (p)\Gamma (n-p+1)}{q}_p^2 \\ +2\ln \left[\iint \frac{dxd\eta }{2\pi }(-i\eta {)}^n\exp \left(i\eta x+\sum _{p=1}^{\infty }\frac{x^p{q}_p}{p!}\right)\right] \end{gathered}$$

$$\frac{\partial \Phi (n,\beta )}{\partial q_p}=0p\in \text{mathds}N.$$

$$\mathcal{C}=\underset{\beta \to +\infty }{\lim }\underset{n\to 0}{\lim }\Phi (n,\beta ).$$

$$-\frac{\ln {\rho }_N(C)}{N}=L(C)+O\left(\frac{1}{N}\right),$$

$$\begin{gathered} \alpha \Phi (\alpha ):=\underset{\begin{array}{c}\beta \to +\infty \\ n\beta =\alpha \end{array}}{\lim }n\beta \Phi (n,\beta )=-\underset{N\to +\infty }{\lim }\frac{\ln \mathbb{E}\left[{\mathrm{e}}^{-\alpha N{C}_N}\right]}{N} \\ =\underset{N\to +\infty }{\lim }\frac{1}{N}\sum _{k=1}^{\infty }(-1{)}^{k-1}\frac{{\kappa }_k{\alpha }^k}{k!}, \end{gathered}$$

$$\begin{gathered} L(C):=-\underset{N\to +\infty }{\lim }\frac{\ln {\varrho }_N(C)}{N}=\underset{\alpha }{\inf }\left[\alpha \Phi (\alpha )-\alpha C\right] \\ ={\alpha }_C\Phi ({\alpha }_C)-{\alpha }_{C}C, \end{gathered}$$

$$C={\frac{\partial \left[\alpha \Phi (\alpha )\right]}{\partial \alpha }|}_{\alpha ={\alpha }_C}.$$

$$\alpha \Phi (\alpha )=\underset{C}{\inf }\left[L(C)+\alpha C\right]=L(C_{\alpha })+\alpha C_{\alpha },$$

$$\alpha ={\frac{\partial L(C)}{\partial C}|}_{C=C_{\alpha }}.$$

$$G_0(t):=\sum _{p=1}^{\infty }\frac{(-1{)}^{p-1}{\mathrm{e}}^{\beta pt}}{p!}{Q}_p,$$

$$G_0(x)=2{\int }_{-x}^{\infty }{\mathrm{e}}^{-G_0(t)}\mathrm{d}t\Rightarrow G_0(x)=\ln \left(1+{\mathrm{e}}^{2x}\right).$$

$$\begin{gathered} \iint \frac{\mathrm{d}x\mathrm{d}\eta }{2\pi }(-i\eta {)}^n{\mathrm{e}}^{i\eta x+{\sum }_{p=1}^{\infty }\frac{x^p{Q}_p}{p!}}= \\ =1+\sum _{k=1}^{\infty }\frac{{\alpha }^k}{{\beta }^{k}k!}\iint \frac{\mathrm{d}x\mathrm{d}\eta }{2\pi }{\ln }^k(-i\eta ){\mathrm{e}}^{i\eta x+{\sum }_{p=1}^{\infty }\frac{x^p{Q}_p}{p!}}. \end{gathered}$$

$$\ln (x)={\int }_0^{+\infty }\frac{{\mathrm{e}}^{-t}-{\mathrm{e}}^{-xt}}{t}\mathrm{d}t$$

$$\begin{gathered} \underset{\begin{array}{c}\beta \to +\infty \\ \beta n=\alpha \end{array}}{\lim }\iint \frac{\mathrm{d}x\mathrm{d}\eta }{2\pi }(-i\eta {)}^n{\mathrm{e}}^{i\eta x+{\sum }_{p=1}^{\infty }\frac{x^p{Q}_p}{p!}}= \\ =1+\alpha \int \left[\theta (-h)-{\mathrm{e}}^{-G_0\left(h\right)}\right]\mathrm{d}h \\ +\frac{{\alpha }^2}{2}\iint \mathrm{d}{h}_1\mathrm{d}{h}_2[\theta (-h_1)\theta (-h_2)-\theta (-h_1){\mathrm{e}}^{-G_0(h_2)} \\ -\theta (-h_2){\mathrm{e}}^{-G_0(h_1)}+{\mathrm{e}}^{-G_0\left(\max \{h_1,h_2\}\right)}]+o({\alpha }^2) \\ =1+\frac{\zeta (2)}{4}{\alpha }^2+o({\alpha }^2). \end{gathered}$$

$$\begin{gathered} \frac{\beta }{2}\sum _{p=1}^{\infty }\frac{\Gamma (n+1)}{\Gamma (p)\Gamma (n-p+1)}{Q}_p^2= \\ =\frac{\alpha }{2}\sum _{p=1}^{\infty }(-1{)}^{p-1}\left[1-\frac{\alpha }{\beta }{\mathrm{H}}_{p-1}+o\left(\frac{\alpha }{\beta }\right)\right]Q_p^2 \\ =\frac{\alpha }{2}\zeta (2)-\frac{{\alpha }^2}{2\beta }\sum _{p=1}^{\infty }(-1{)}^{p-1}{\mathrm{H}}_{p-1}{Q}_p^2+o({\alpha }^2). \end{gathered}$$

$$\begin{gathered} \underset{\begin{array}{c}\beta \to +\infty \\ n\beta =\alpha \end{array}}{\lim }\frac{\beta }{2}\sum _{p=1}^{\infty }\frac{\Gamma (n+1)}{\Gamma (p)\Gamma (n-p+1)}{Q}_p^2= \\ =\alpha \frac{\zeta (2)}{2}+{\alpha }^2{\int }_{-\infty }^{+\infty }\mathrm{d}h{\mathrm{e}}^{-G_0(h)}{\int }_0^{+\infty }{G}_0(h-t)\mathrm{d}t+o({\alpha }^2) \\ =\alpha \frac{\zeta (2)}{2}+{\alpha }^2\frac{\zeta (3)}{2}+o({\alpha }^2). \end{gathered}$$

$$\Phi (\alpha )=\frac{\zeta (2)}{2}-\frac{\zeta (2)-\zeta (3)}{2}\alpha +o(\alpha ),$$

$$\mathbb{E}\left[(C_N-\mathcal{C}{)}^2\right]=\frac{\zeta (2)-\zeta (3)}{N}+o\left(\frac{1}{N}\right).$$

$$\int (-i\eta {)}^n{\mathrm{e}}^{i\eta x}\mathrm{d}\eta =-\frac{2\Gamma (n+1)\sin (n\pi )}{(-x{)}^{n+1}}\theta (-x),$$

$$G_{n,\beta }(t):=\sum _{p=1}^{\infty }\frac{(-1{)}^{p-1}{Q}_p{\mathrm{e}}^{\beta pt}}{p!}$$

$$\begin{gathered} \iint \frac{\mathrm{d}x\mathrm{d}\eta }{2\pi }(-i\eta {)}^n\exp \left(i\eta x+\sum _{p=1}^{\infty }\frac{x^p{Q}_p}{p!}\right)= \\ =-\frac{\beta \Gamma (n+1)\sin (n\pi )}{\pi }{\int }_{-\infty }^0\exp \left(\sum _{p=1}^{\infty }\frac{x^p{Q}_p}{p!}\right)\frac{\mathrm{d}x}{(-x{)}^{n+1}} \\ =-\frac{\beta \Gamma (n+1)\sin (n\pi )}{\pi }{\int }_{-\infty }^{+\infty }{\mathrm{e}}^{-n\beta t-G_{n,\beta }(t)}\mathrm{d}t. \end{gathered}$$

$$\begin{gathered} \beta p\left(\genfrac{}{}{0ex}{}{n}{p}\right)Q_p=2\frac{(-1{)}^p}{p!}\frac{{\int }_{-\infty }^{+\infty }{\mathrm{e}}^{(p-n)\beta t-G_{n,\beta }(t)}\mathrm{d}t}{{\int }_{-\infty }^{+\infty }{\mathrm{e}}^{-n\beta t^{\prime }-G_{n,\beta }(t^{\prime })}\mathrm{d}{t}^{\prime }}⟹ \\ {G}_{n,\beta }(x)=-2\frac{\underset{-\infty }{\overset{+\infty }{\int }}{K}_{n,\beta }(x+t){\mathrm{e}}^{-n\beta t-G_{n,\beta }(t)}\mathrm{d}t}{{\int }_{-\infty }^{+\infty }{\mathrm{e}}^{-n\beta t^{\prime }-G_{n,\beta }(t^{\prime })}\mathrm{d}{t}^{\prime }}. \end{gathered}$$

$$K_{n,\beta }(u):=\sum _{p=1}^{\infty }\frac{{\mathrm{e}}^{\beta up}}{\left(\genfrac{}{}{0ex}{}{n}{p}\right)\beta p(p!{)}^2}$$

$$\underset{\begin{array}{c}\beta \to +\infty \\ n\beta =\alpha \end{array}}{\lim }{K}_{n,\beta }(u)=\theta (u)\left(\frac{1}{\alpha }-u\right),$$

$$\begin{gathered} G_{\alpha }(x):=\underset{\begin{array}{c}\beta \to +\infty \\ n\beta =\alpha \end{array}}{\lim }{G}_{n,\beta }(x) \\ =-\frac{2\alpha }{{\mathcal{Z}}_{\alpha }}{\int }_{-x}^{+\infty }\left(x+t-\frac{1}{\alpha }\right){\mathrm{e}}^{-\alpha t-G_{\alpha }(t)}\mathrm{d}t, \end{gathered}$$

$${\mathcal{Z}}_{\alpha }:=-\alpha {\int }_{-\infty }^{+\infty }{\mathrm{e}}^{-\alpha t^{\prime }-G_{\alpha }(t^{\prime })}\mathrm{d}{t}^{\prime }.$$

$$\alpha \Phi (\alpha )=\frac{\alpha }{{\mathcal{Z}}_{\alpha }}{\int }_{-\infty }^{+\infty }{G}_{\alpha }(t){\mathrm{e}}^{-\alpha t-G_{\alpha }(t)}\mathrm{d}t-2\ln {\mathcal{Z}}_{\alpha }.$$

$$P(m_{ij})=\frac{\exp \left[-\beta m_{ij}(N{w}_{ij}-{\varphi }_i-{\varphi }_j)\right]}{1+\exp \left[-\beta (N{w}_{ij}-{\varphi }_i-{\varphi }_j)\right]}.$$

Full text

I Introduction

The application of replica theory Mézard et al. (1987) to the study of random combinatorial optimization problems (RCOPs) has a long tradition that started more than thirty years ago with the seminal works by Orland (1985) and Mézard and Parisi (1985). It became immediately clear that methods borrowed from the theory of disordered systems are very effective to study the average properties of ensembles of RCOPs. Exact results have been obtained, e.g., for mean-field versions of many RCOPs, such as the matching problem Mézard and Parisi (1985); *Mezard1988; *Lucibello2017; *Lucibello2018, the travelling salesman problem Mézard and Parisi (1986a); *Krauth1989a, $K$-SAT problems Mézard et al. (2002); *Mezard2002b; *Braunstein2007; *Krzakala2007, graph partitioning Fu and Anderson (1986); *Kanter1987, matching enumeration in sparse graphs Zdeborová and Mézard (2006), constraint least square problems Fyodorov and Le Doussal (2014), and many others. In the case of the matching problem, expressions for the finite-size corrections to the average optimal cost have also been obtained Mézard and Parisi (1987); *Parisi2002; *Caracciolo2017. The parallel success of the cavity method Mézard et al. (1987); Mézard and Parisi (2001); *Mezard2003 inspired message-passing algorithms for the solution of specific instances of RCOPs Mézard and Montanari (2009).

In this paper, we study the deviations from the average optimal cost (AOC) in a particular RCOP, the random-link matching problem, using both the replica and the cavity method. Such an investigation is of methodological interest beyond the analysis of the specific problem. Replica calculations for the study of large deviations of thermodynamic-like functionals in presence of disorder are quite rare in the literature. Indeed, except for the analysis in Ref. Fyodorov and Le Doussal (2014), where the replica method is applied to the study of large deviations of the minimum, the results cited above mainly concern the typical properties of the solutions only. On the other hand, the application of the cavity method to the study of large fluctuations has not been explored up to now, with the exception of Ref. Rivoire (2005). Here we apply, for the first time, both approaches to the same problem, showing that they lead to the same result.

We will study both the small fluctuations around the AOC and the large deviations from it. In the matching problem we assume that $2N$ vertices, labelled by the index $i\in\{1,\dots,2N\}$ are given, alongside with a positive weight $w_{ij}\in\mathds{R}^{+}$ for each pair $(i,j)$. We search for the symmetric matrix $\mathbf{M}=(m_{ij})_{i,j}$ such that the cost

[TABLE]

is minimized. The minimization has to be performed on the set of matrices $\mathbf{M}$ such that

[TABLE]

In the random-link version of the problem, the quantities $w_{ij}$ are supposed to be i.i.d. random variables, with probability density function $\rho(w)$. Using the replica theory, in Ref. Mézard and Parisi (1985) it has been proven that, if $\lim_{w\to 0}\rho(w)=1$, then colorblue

[TABLE]

where we have denoted the average over all possible realizations by ${\mathbb{E}\left[\bullet\right]}$, and $\zeta(z)$ is the Riemann zeta function. The calculation was performed introducing a partition function

[TABLE]

where the indicator function $\mathbb{I}(\bullet)$ is equal to one if its argument is true, zero otherwise. From the expression above, the replicated free-energy can be derived

[TABLE]

The functional $\Phi(n,\beta)$ has been obtained, in the replica symmetric hypothesis, in Ref. Mézard and Parisi (1985) and it is equal to

[TABLE]

The asymptotic AOC is then recovered as the value of $\Phi$ in the zero temperature limit, taking the number of replicas $n$ going to zero,

[TABLE]

Observe that we should, in principle, take $n\to 0$ first, and then $\beta\to+\infty$, but, as usual in replica calculations, we will assume that the order of two limits can be safely inverted. Here and in the following, we denote by $C_{N}\coloneqq\min_{\mathbf{M}}C_{N}[\mathbf{M}]$ the instance-dependent optimal cost, and by $\varrho_{N}(C)$ its distribution. We expect $\rho_{N}(C)$ to concentrate, for $N\to+\infty$, on the asymptotic AOC $\mathcal{C}$, $\lim_{N\to+\infty}\varrho_{N}(C)=\delta(C-\mathcal{C})$. In particular, we will show that the fluctuations around the AOC are Gaussian. Moreover, we expect $C_{N}$ to satisfy a large deviation principle, i.e.,

[TABLE]

where $L(C)$ is a strictly convex, positive function having $L(\mathcal{C})=0$, known as the Cramér rate function. In some sense, $L(C)$ measures how rare is to find an optimal cost $C\neq\mathcal{C}$ for large values of $N$.

Our computation of the large deviation function $L(C)$ for the random-link matching problem starts exactly from Eq. (6a). It is well known, indeed, that the replicated average free-energy $n\Phi(n,\beta)$ contains information not only on the average free energy, but also on its fluctuations Crisanti and Leuzzi (2014). In particular, using Eq. (5) it is possible to show that, for finite $n$, $-n\beta\Phi(n,\beta)$ is the cumulant generating function for the free energy $\ln Z_{w}(\beta)$ Crisanti et al. (1992). This fact has been used, for example, by Parisi and Rizzo Parisi and Rizzo (2008); *Parisi2009; *Parisi2010a, to extract the large deviation function in the Sherrington-Kirkpatrick model. Thet confirmed the anomalous scaling of fluctuations of the free energy in the RSB phase predicted, near the critical temperature, by Crisanti et al. (1992), on the basis of a previous result by Kondor (1983). In the present paper, we are interested in the fluctuation of the “ground state free-energy” $C_{N}$ in the random-link matching problem, and therefore we have to take $\beta\to+\infty$. The cumulant generating function of the optimal cost is then obtained as

[TABLE]

where $\kappa_{k}$ is the $k$-th cumulant of the random variable $NC_{N}=\lim_{\beta\to+\infty}\beta^{-1}\ln Z_{w}(\beta)$ Crisanti et al. (1992), the first order in $\alpha$ simply being the asymptotic AOC, $\alpha\Phi(\alpha)=\alpha\mathcal{C}+o(\alpha)$. If $\lim_{N}N^{-1}\kappa_{2}$ is finite and different from zero, i.e., $\kappa_{2}=2\sigma^{2}N$ with $\sigma^{2}=O(1)$, this implies that ${\mathbb{E}\left[(C_{N}-\mathcal{C})^{2}\right]}=\sigma^{2}N^{-1}$, i.e., small fluctuations of the optimal cost are Gaussian.

If $\alpha\Phi(\alpha)$ is differentiable and convex, by the Gärtner–Ellis theorem the Cramér function of $\varrho_{N}(C)$ is obtained as the Legendre-Fenchel transform of $\alpha\Phi(\alpha)$,

[TABLE]

Vice-versa, the cumulant generating function can be obtained from the rate function using, once again, a Legendre-Fenchel transform,

[TABLE]

Intuitively, therefore, the quantity $\alpha^{-1}$ plays the role of a “fluctuation scale” with respect to the deviation $C-\mathcal{C}$. Indeed, assuming $L(C)$ strictly convex and nonnegative, such that $L(\mathcal{C})=0$, then $\alpha=0$ corresponds to $C_{\alpha}=\mathcal{C}$. On the other hand, being $\partial_{C}L(C)<0$ for $C<\mathcal{C}$, it is clear that positive values of $\alpha$ corresponds to negative fluctuations of the cost, and negative values of $\alpha$ corresponds to positive deviations from the asymptotic AOC.

In the following, we will derive the exact value of $\sigma^{2}$, proving that the small fluctuation of the optimal cost are indeed Gaussian, and we will obtain an expression for the function $\Phi(\alpha)$ that we will solve numerically.

II Small fluctuations

Let us start from the computation of the variance of the optimal cost. The small $\alpha$ expansion of $\alpha\Phi(\alpha)$ up to $o(\alpha^{2})$ terms will provide us the first and the second cumulant of the optimal cost, i.e., its mean and its variance. Due to the fact that we are performing an expansion around $\alpha=0$, it is useful to recall the expression of the saddle-point value of the order parameter $q_{p}\equiv Q_{p}$ in this particular case, i.e., for $n\to 0$ and $\beta\to+\infty$. It has been shown in Ref. Mézard and Parisi (1985) that, introducing the function

[TABLE]

the saddle-point condition for $n\to 0$ and $\beta\to+\infty$ reads

[TABLE]

To obtain the expansion of $\Phi(\alpha)$, let us start from the double integral appearing in the argument of the logarithm in Eq. (6a), to be evaluated on the saddle-point $Q_{p}$ for small $\alpha$. We have

[TABLE]

Using now the representation of the logarithm

[TABLE]

we can write

[TABLE]

Similarly, the first term in Eq (6a) can be written as

[TABLE]

where $\mathrm{H}_{p}$ is the $p$th harmonic number. If we now use the fact that $\mathrm{H}_{p-1}=\int_{0}^{\infty}\frac{\mathrm{e}^{-t}-\mathrm{e}^{-pt}}{1-\mathrm{e}^{-t}}\mathrm{d}t$, then

[TABLE]

Collecting all results we get

[TABLE]

implying that small fluctuations of the optimal cost in the random-link matching problem are Gaussian, with a variance given by

[TABLE]

Recently Wästlund (2010) has derived the variance of the random-link assignment problem with exponentially distributed random weights on a complete bipartite graph. His result, obtained using a purely probabilistic approach, coincides with Eq. (19), apart from a global factor $4$, due to the fact that the optimal cost of the assignment problem is twice the optimal cost of the matching problem for $N\to+\infty$. In our calculation, only the assumption $\lim_{w\to 0}\rho(w)=1$ for the weight probability density function has been used.

III Large deviations via replicas

In the previous Section we have performed a small $\alpha$ expansion to extract the variance of the optimal cost $C_{N}$ for $N\gg 1$. To get instead the large deviation function $L(C)$, we have to keep $\alpha$ finite. We follow two different approaches and we compare then our results with numerical simulations. Let us start from the replica method. Following our general recipe, we derive the large deviation function starting from Eq. (6a), writing down the saddle-point equation in the $\beta\to+\infty$ limit, taking $n\beta=\alpha$ fixed. We start considering $\alpha<0$ (we will relax this assumption in our cavity calculation). Using the fact that, for $n\in(-1,0)$,

[TABLE]

and defining as in Eq. (12)

[TABLE]

the argument of the logarithm can be re-written as

[TABLE]

The saddle-point equation (6b) becomes

[TABLE]

where the function

[TABLE]

appears. Observing now that (see Appendix)

[TABLE]

the saddle-point equation can be written as

[TABLE]

Eq. (26a) can be solved numerically for a given value of $\alpha$, allowing then to evaluate $\alpha\Phi(\alpha)$ in Eq. (26c), whose Legendre transform is the desired large deviation function $L(C)$. Using the properties of $G_{\alpha}$ derived above, it can be seen that $\lim_{\alpha\to-2^{+}}\alpha\Phi(\alpha)=-\infty$: the presence of such a singularity gives us information on the large $C$ behavior of the Cramér function, i.e., it implies that $\lim_{C\to+\infty}C^{-1}L(C)=2$. As anticipated, Eqs. (26) have been derived assuming $-2<\alpha<0$. To get an expression that can be prolonged to positive values of $\alpha$ we will use the cavity method.

IV Large deviations via cavity

The equation for $\Phi(\alpha)$ given by the replica method for $\alpha\in(-2,0]$ can be also obtained using the cavity method and actually extended to positive values of $\alpha$. The starting point is the cavity condition for the occupancy of an edge in the random-link matching problem. In particular, in the cavity approach, each edge $(i,j)$ is associated to its weight $w_{ij}$ and to two cavity fields, $\phi_{i}$ and $\phi_{j}$ on its vertices, containing information on the rest of the graph, in such a way that the occupancy $m_{ij}$ of the the edge is distributed as

[TABLE]

In the $\beta\to+\infty$ limit, an edge is occupied if, and only if, $Nw_{ij}<\phi_{i}+\phi_{j}$. At zero temperature and in the large $N$ limit, the cavity fields satisfy the following equation Mézard and Parisi (1986b); Krauth and Mézard (1989); Parisi and Ratiéville (2001)

[TABLE]

Here $\partial 0$ is the set of neighbors of the node [math]. The mate node $i^{*}$ of [math] is such that

[TABLE]

In Refs. Mézard and Parisi (1986b); Krauth and Mézard (1989); Parisi and Ratiéville (2001) the previous equations have been studied and solved. The AOC predicted by the cavity method coincides with the one obtained using the replica approach.

The recurrence relation for the cavity fields can be also used to extract information on the fluctuations and evaluate $\alpha\Phi(\alpha)$. For the sake of simplicity, let us start from a different version of the problem, i.e., the random-link matching problem on a sparse graph, and in particular on a Bethe lattice, and let us follow the approach of Rivoire (2005) for the study of large deviation on sparse topologies. In this case, we are interested in solving our problem on a graph having $2N$ vertices, each one of them having coordination $z$: we will later take the limit $z\to 2N-1$. Taking this limit might sound dangerous, because we apply a result obtained for a sparse topology to a dense one. However, the random-link matching problem is an “effectively sparse” problem: given a fully-connected topology, the probability that a given node is connected, in the optimal matching, to its $n$th nearest neighbor is exponentially small in $n$ Parisi and Ratiéville (2001). We will denote by $L_{z}(C)$ the large deviation function for random-link matching problem on the Bethe lattice, so that $L(C)=\lim_{z\to+\infty}L_{z}(C)$.

To obtain an expression for it, we proceed as usual in the cavity approach, i.e., starting from an intermediate graph having $z$ randomly chosen (cavity) nodes with coordination $z-1$, and all the other nodes with coordination $z$.

Let us denote by $\hat{L}_{z}$ the large deviation function corresponding to the random-link matching problem on such a topology.

We can recover the “correct” Bethe lattice topology in two ways. We can, for example, connect a new node to the $z$ cavity nodes.

The optimal cost will be shifted by a certain amount $N^{-1}\varepsilon$, in such a way that the probability density function of the optimal cost satisfies the equation

[TABLE]

where $\alpha\coloneqq-\partial_{C}\hat{L}_{z}(C)$ and $p_{v}(\varepsilon)$ is the distribution of the energy-shift $\varepsilon$ due to a vertex addition.

Another possibility is to add $\nicefrac{{z}}{{2}}$ edges.

We obtain in this case the following relation

[TABLE]

where $p_{e}(\epsilon)$ is the distribution of the energy-shift $\varepsilon$ due to an edge addition. Taking the ratio of the two expressions above, we obtain an equation for $L_{z}(C)$ at the leading order in $N$, namely

[TABLE]

Taking $z=2N-1\approx 2N$ we obtain the expression for our case. To evaluate the previous quantity, let us introduce the joint distribution $p_{v}(\phi,\varepsilon)$ of the cavity field entering in the added node and of the energy shift, such that $p_{v}(\varepsilon)=\int p_{v}(\phi,\varepsilon)\mathrm{d}\phi$. We define the reweighted distribution of the cavity field

[TABLE]

with $\mathcal{Z}_{\alpha}$ proper normalization constant. In our case

[TABLE]

because the cost shift due to the addition of a node coincides with the incoming cavity field, see Eq. (28). If we denote by

[TABLE]

the distribution of the cavity field can be rewritten as

[TABLE]

where we have used Eq. (28) to express $p_{v}$ in terms of $\pi_{\alpha}$. In the $N\to+\infty$ limit we obtain an equation for $p_{\alpha}$,

[TABLE]

Moreover, because of Eq. (27), the energy cost due to the addition of an edge

[TABLE]

and therefore, by means of an integration by parts, we get for large $N$

[TABLE]

Note that, up to now, no assumptions have been made on the range of values of $\alpha$, except the implicit ones about the fact that the quantities above are well defined and convergent: the cavity expression can be used therefore for both positive and negative values of $\alpha$, provided that the involved quantities are finite.

It can be seen that the expression in Eq. (37) is equivalent to the one given in Eq. (26c) in its range of validity. Indeed, introducing the function

[TABLE]

Eq. (35) simplifies as

[TABLE]

and therefore we can write, using Eq. (38), a self-consistent equation for the function $G_{\alpha}(\phi)$ that is found to be identical to Eq. (26a), proving that the function $G_{\alpha}$ introduced here is the same appearing in the replica approach. Repeating the arguments presented in the replica derivation, we obtain that the integrals are finite for $\alpha>-2$. Substituting Eq. (39) in Eq. (37), simple manipulations give us the same expression presented in Eq. (26c), the main difference being that, in the obtained formula,

[TABLE]

If we further restrict ourselves to negative values of $\alpha$, an integration by parts allows us to write $\mathcal{Z}_{\alpha}$ in the same form given in Eq. (26a), proving the equivalence of the two approaches and implicitly providing us a prolongation of the replica result.

V Numerical results

We have integrated Eq. (37) by means of a population dynamics algorithm, and we have compared our results with the value of $\alpha\Phi(\alpha)$ obtained by numerical simulations, performed solving a large number of instances of the problem. In our numerical simulations, we assumed $\rho(w)=\mathrm{e}^{-w}$. The quantity $\alpha\Phi(\alpha)$ can be evaluated directly, using the fact that $\alpha\Phi(\alpha)=\lim_{N\to+\infty}N^{-1}\ln\mathbb{E}\left[\mathrm{e}^{-\alpha NC_{N}}\right]$. The agreement is very good in the neighborhood of the origin, as it can be seen in Fig. 1, where the cavity and numerical results are compared. For reference, we have plotted the tangent to the curve in the origin, whose slope coincides with the AOC, $\nicefrac{{\zeta(2)}}{{2}}$. We have discarded the data points where finite-sample effects appear for larger values of $|\alpha|$; obviously, better estimates can be obtained running an exponentially large number of instances in the size of the system. Finite-size effects still appear in the evaluation of the derivative of $\alpha\Phi(\alpha)$ (that has been also directly evaluated as $\partial_{\alpha}[\alpha\Phi(\alpha)]=\lim_{N\to+\infty}\frac{\mathbb{E}[C_{N}\mathrm{e}^{-\alpha NC_{N}}]}{\mathbb{E}[\mathrm{e}^{-\alpha NC_{N}}]}$) for $\alpha<0$, larger sizes being closer to the theoretical predictions, see Fig. 2. Finally, the theoretical prediction near $\alpha=-2$ becomes noisy and less reliable, because of the approaching of the divergence. In this regime, the convergence of the cavity fields distribution fails and an accurate estimation of $\alpha\Phi(\alpha)$ is more challenging. In Fig. 3 we have plotted the Legendre transform of our cavity results, that is the desired large deviation function. The curve is compared with the quadratic approximation in the origin, corresponding to the Gaussian small-fluctuation behavior.

VI Conclusions

Using the replica approach we have evaluated the variance of the average optimal cost for the random-link matching problem on the complete graph, assuming a distribution of the weights such that $\lim_{w\to 0}\rho(w)=1$. Our result is in agreement with a previously obtained expression by Wästlund, proving that the small fluctuations of the optimal cost around its asymptotic AOC are Gaussian. We have then derived an expression for the Legendre transform $\alpha\Phi(\alpha)$ of the Cramér function $L(C)$ using both the replica theory (for positive cost fluctuation) and the cavity method. The cavity formula, in particular, has been obtained for the random-link matching problem on a generic sparse graph having fixed coordination, and provides a recipe for the numerical evaluation of $\alpha\Phi(\alpha)$. In the fully-connected case, our results also show that $\alpha\Phi(\alpha)$ diverges for $\alpha\to-2$, implying that $\lim_{C\to+\infty}C^{-1}L(C)=2$.

Acknowledgments

We thank Johan Wästlund for communicating us his results. GS also thanks Sergio Caracciolo and Yan Fyodorov for stimulating the study of large deviations in the random-link matching problem. We thank Federico Ricci-Tersenghi and Tommaso Rizzo for useful discussions. GP and GS acknowledge financial support from the Simons Foundation (grant No. 454949, G. Parisi).

Appendix A Derivation of Eq. (25)

To prove the limit in Eq. (25), we can start using the fact that $\binom{n}{k}=(-1)^{k}\binom{-n+k-1}{k}$. The series can be written as

[TABLE]

We have to compute the $\beta\to+\infty$ limit at $n\beta=\alpha$ fixed. The coefficient in front of the series has limit $\lim_{\beta\to+\infty}\beta^{-1}\Gamma\left(-\nicefrac{{\alpha}}{{\beta}}\right)=-\nicefrac{{1}}{{\alpha}}$. On the other hand,

[TABLE]

where we have used the fact that

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We can write the following expansion

[TABLE]

where we have introduced the (opposite) polygamma function $\hat{\psi}_{n}(z)\coloneqq-\psi_{n}(z)$

[TABLE]

and the (complete) exponential Bell polynomials

[TABLE]

The expansion above is obtained applying the Faà di Bruno formula for the derivative of a composed function $\partial^{n}_{x}(f\circ g)(x)$ with $f(x)\equiv\mathrm{e}^{x}$ and $g(x)\equiv-\ln\Gamma(x)$. If we now assume $\mathrm{B}_{0}\equiv 1$, the Bell polynomials satisfy the recurrence relations Charalambides (2002)

[TABLE]

We will show now by induction that, for $k\geq 2$,

[TABLE]

For $k=1$ we have

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The $k=2$ case follows straightforwardly. Indeed,

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Let us suppose now that the statement is true for generic $k>2$ and let us consider it for $k+1$. Then we have

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The inner sum can be written as

[TABLE]

implying that

[TABLE]

The last two terms in the previous expression in the $\beta\to\infty$ tend to zero

[TABLE]

By the induction hypothesis, the remaining $k-2$ contributions are infinitesimal as well for $\beta\to+\infty$, and the thesis is proved. We can restrict therefore the expansion in Eq. (44) to the $k=1$ term in the $\beta\to+\infty$ hypothesis. We have

[TABLE]

The relations above finally give us the asymptotic in Eq. (25).