Zipfs Law from Scale-free Geometry

Zipf ’s La w from Scale-free Geometry Henry W. Lin 1 and Abraham Lo eb 2 1 Harvar d Col le ge, Cambridge, MA 02138, USA 2 Institute for The ory & Computation, Harvar d-Smithsonian Center for Astr ophysics, 60 Gar den Stre et, Cambridge, MA 02138, USA (Dated: F ebruary 16, 2016) The spatial distribution of people exhibits clustering across a wide range of scales, from household ( ∼ 10 − 2 km) to con tinental ( ∼ 10 4 km) scales. Empirical data indicates simple p ow er-law scalings for the size distribution of cities (known as Zipf ’s la w) and the p opulation density fluctuations as a function of scale. Using techniques from random field theory and statistical physics, we show that these pow er laws are fundamen tally a consequence of the scale-free spatial clustering of h uman popu- lations and the fact that humans inhabit a t w o-dimensional surface. In this sense, the symmetries of scale inv ariance in tw o spatial dimensions are intimately connected to urban so ciology . W e test our theory by empirically measuring the p ow er sp ectrum of p opulation density fluctuations and show that the logarithmic slop e α = 2 . 04 ± 0 . 09, in excellent agreemen t with our theoretical prediction α = 2. The mo del enables the analytic computation of man y new predictions by imp orting the mathematical formalism of random fields. I. INTR ODUCTION Human p opulations exhibit remark ably simple prop er- ties given the complexity of so cio economic interactions b et ween h umans and their environmen ts[1]. One such example is the well known Zipf ’s law [2] for cities: the rank of a city is inv ersely prop ortional to the n umber of p eople who live in the city . If the most p opulous city in the United States has a p opulation of N max,US ∼ 8 × 10 6 , the second most p opulous cit y will hav e a p opulation of 1 2 N max,US ∼ 4 × 10 6 , the third 1 3 N max,US ∼ 2 . 7 × 10 6 , and so forth. This simple relation fits empirical data extremely w ell[3, 4]. A mathematically equiv alent for- m ulation of Zipf ’s law is that the underlying distribution of cities follo ws a p ow er law[5]; namely , the probabilit y that a city has a p opulation N scales as 1 / N 2 . The remark able simplicity and empirical success of Zipf ’s law ha ve attracted significan t theoretical atten tion and debate[3, 6, 7], though there is no consensus on the origin of Zipf ’s law. Existing work treats cities as the fundamen tal entities of the theory , with p opulation as a prop ert y of each city . F or example, Gibrat’s law applied to cities[3, 8, 9], which states that the fractional growth rate of a city is indep endent of its p opulation, will drive the distribution of cit y p opulations to a log-normal dis- tribution. The tail of the log-normal distribution then giv es rise Zipf ’s law. Our approach is conceptually differen t: w e treat the p opulation densit y as the fundamental quantit y , thinking of cities as ob jects that form when the p opulation den- sit y exceeds a critical threshold. The situation is there- fore conceptually and mathematically analogous to the formation of galaxies in the univ erse, where non-linear gra vitational collapse occurs when the matter density ex- ceed some critical v alue. Our conceptual adv ance here is also a practical one, since we can apply the mathemati- cal to ols developed for analyzing random fields[10] to the problem at hand. Before pro ceeding with a technical deriv ation of our re- sults, let us briefly summarize them. The starting p oint is to mo del human p opulation density as a random func- tion of spatial p osition. A function of spatial p osition is a field , and thus human p opulation density will b e mo d- eled as a r andom field (for a review of relev an t topics in random fields, see [11, 12] and esp ecially [13]). T o lo west order, a single random v ariable in elemen tary statistics is c haracterized by a mean and a v ariance. A random field ma y b e regarded as a higher dimensional generalization of a single random v ariable. By analogy , a random field is characterized b y a mean and a p ower sp e ctrum , which can b e thought of as a generalization of v ariance. The p o wer sp ectrum gives the amount of fluctuations of the field as a function of scale. T o derive the form of the p o wer spectrum for human population density , w e in- v oke scale-inv ariant random growth, similar in spirit to Gibrat’s law. W e mov e on to derive Zipf ’s la w. Our deriv ations in volv e the simple assumption that some cities emerge ab o ve some critical p opulation densit y threshold. T o coun t the num b er of cities in our mo del, one must an- sw er the following mathematical question: giv en a ran- dom field characterized by a p ow er sp ectrum, how often do es the random field take on v alues greater than a cer- tain threshold? This is a frequently asked question in the context of cosmology , and the Press-Schec h ter (PS) formalism allows us to analytically compute the answer. W e demonstrate that our deriv ation of Zipf ’s law is more general than the motiv ating random growth model; w e argue that the only key ingredient is scale inv ariance in t w o spatial dimensions. In other w ords, whereas previ- ous work tends to fo cus on how Zipf ’s law emerges from concrete mo dels, we argue that Zipf ’s la w naturally o c- curs in a v ery large class of statistical models. In the lan- guage of statistical physics[12], the existence of Zipf ’s la w is only a function of the universalit y class of the statisti- cal mo del; it is indep endent of the “microscopic” details of the system’s dynamics whic h are undoubtedly complex in the case of human p opulations. 2 I I. DERIV A TION OF ZIPF’S LA W W e no w proceed with the detailed deriv ation. T o start, consider the human p opulation density ρ as a function on R 2 , the 2D Euclidean plane. Since w e will b e inter- ested in regions muc h smaller in size than the radius of the Earth, w e will ignore the effects of curv ature. The fluctuations relativ e to the a v erage p opulation densit y δ ( x ) ≡ [( ρ ( x ) / ¯ ρ ) − 1] can b e expanded in F ourier mo des δ ( x ) = 1 2 π Z d 2 k δ k e − i kx . Up to a con ven tional normalization factor of 2 π , this equation simply rewrites the population fluctuations as a sum of plane w av es e − i kx , each w eighted by a factor δ k . Since the left hand side is a random v ariable, the right hand side must also b e a random v ariable; since every term except for δ k on the right hand side is manifestly deterministic, δ k m ust b e a contin uum of random v ari- ables, with one random v ariable for each wa ve vector k . Just as an ordinary random v ariable is c haracterized b y a v ariance, each δ k is characterized by a num b er P ( k ) called the p ow er sp ectrum, whic h is defined as h δ k δ ∗ k 0 i = (2 π ) 2 δ 2 D ( k − k 0 ) P ( k ) , (1) where δ D is the Dirac delta function (not to b e confused with the fractional ov er-densit y δ ( x ). By assuming rota- tional symmetry , the p ow er spectrum becomes a function only of magnitude P ( k ) = P ( k ). Equation (1) makes pre- cise the statement that the p ow er sp ectrum P ( k ) quanti- fies the amount of statistical fluctuations asso ciated with a given frequency k . It is conv en tional to define a dimensionless pow er spec- trum in the n umber densit y ∆ 2 ( k ) ≡ k 2 P ( k ) / (2 π ), whic h represen ts the typical (squared) fractional ov er-density of p eople ( δ ρ/ρ ) 2 on the spatial scale ∼ 1 /k . T o make fur- ther progress, w e must fix the functional form of ∆( k ) b y some theoretical principle. T o this end, consider an o ver-densit y of size ∼ 1 /k . A t a discrete time step, this o ver-densit y migh t gro w or shrink in spatial co v erage. As a concrete example, consider a collection of farms (with a c haracteristic p opulation density of a few p eople p er typ- ical farm area) in otherwise relativ ely uninhabited coun- tryside. A t eac h time step, a farm could b e added or destro yed. In this wa y , our unifying principle of random w alkers is conceptually similar to previous work on the random growth of firms [14]. Therefore, the spatial size of the o ver-densit y might grow or shrink, while δ ρ/ρ (a num- b er asso ciated with farms) will b e held constant. More precisely , we define a monotonically decreasing function X ( k ) suc h that lim k →∞ X = 0, whic h quantifies the spa- tial extent of an ov er-densit y . This function might rep- resen t the area of the ov er-density X ( k ) ∝ 1 /k 2 or its p erimeter X ( k ) ∝ 1 /k , but our deriv ation will not de- p end on the detailed form of X . W e can then p erform a change of v ariables and view ∆( k ) as a function of X : ∆( X ( k )) = ∆( k ). The unifying principle is that all o ver-densities can grow or shrink spatially , executing a random w alk in X . This pro cess can contin ue un til the o verdensit y disapp ears ( X = 0), or the o v er-density tak es up some maximum X max , where X max ≡ X ( k min ) is set b y the continen tal length scale ∼ 1 /k min . F or a large ensem ble of ov er-densities, this is a diffusion-like pro cess with reflecting b oundary conditions ob eying ∂ ∆ ∂ t = D ∂ 2 ∆ ∂ X 2 (2) with some diffusion constant D . W e are only interested in the late-time b ehavior of equation (2). An y initial conditions will relax to the steady-state solution ∆( X ) → constan t for 0 ≤ X ≤ X max on a timescale T relax ∼ X 2 max /D . W e intuitiv ely exp ect T relax to b e reasonably short, since the geographic mobility timescale of ∼ 5 yrs (in the United States, ∼ 35% of p eople change residences within 5 years[15]) is considerably shorter than, say , the p opulation growth timescale ∼ 30 yrs set by the typical age of parenthoo d. Any initial conditions set by an tiquity or p erturbations to the system (e.g. catastrophic even ts that displace man y p eople) should b e quic kly erased. W e therefore predict that on sufficiently long timescales, P ( k ) ∝ k − 2 . (3) W e test this prediction in Figure 1 against publicly av ail- able data from the Cen ter for In ternational Earth Science Information Netw ork (CIESIN) and Centro Internacional de Agricultura T ropical (CIA T)[16]. W e find the b est fit slop e P ( k ) ∝ k − α to b e α = 2 . 04 ± 0 . 09, where we hav e rep orted the ± 1 σ uncertainties. Our theoretical predic- tion is therefore in excellen t agreemen t with observ ations across a broad range of spatial scales, from a few km to ∼ 10 3 km. Before further developing the theory , a more intuitiv e deriv ation of P ( k ) ∼ k − 2 is worth men tioning. Over a large range of length scales our mo del is scale free, imply- ing P ( k ) ∼ k − α for some α . In d spatial dimensions, the left hand side of equation (1) has units of k − 2 d , the Dirac delta function has units of k − d , so P ( k ) should ha ve units of k − d . Since there are no other dimensional parameters relev ant to our theory (the diffusion constant has units of [ X ] 2 T − 1 , but there are no other constant with units of time T ), we must hav e α = d = 2 in tw o spatial di- mensions. In this sense, geometry and scale in v ariance uniquely determines the slop e of the p o wer sp ectrum. In fact, this simple argument demonstrates that P ( k ) ∝ k − 2 is a universal feature of 2D models that hav e no parameters with units of length to some pow er. Effec- tiv e field theory , a p ow erful technique for studying any statistical ph ysics system, can be used to further sharp en this statement; this is done in App endix D. The deriv a- tion in App endix D pro vides p erhaps the most rigorous w ay of justifying the stateme n t that P ( k ) ∝ k − 2 is a generic prop erty of mo dels with scale inv ariance in tw o dimensions, since effective field theory should capture the equilibrium prop erties of any statistical mo del on scales 3 | δ  | 2 0.5 × 10 - 2 0.02 0.05 0.1 1 10 100 1000 10 4  [  -  ] FIG. 1. (Color online). Empirically measured p ow er sp ec- trum P ( k ) ∝ | δ k | 2 ∼ k − α of p opulation density fluctuations as a function of the spatial wa v enum ber k . The b est fit slop e α = − 2 . 04 ± 0 . 09 (solid blue line) is virtually indistinguishable from the predicted slop e α = − 2 (dashed orange line). The data was obtained by taking the diagonal entries (to av oid anisotrop y from rectangular gridding) of a discrete F ourier transform of a 1000 × 1000 arcmin 2 map of the p opulation densit y of a section of the continen tal United States. The area was selected to minimize artifacts due to b oundary con- ditions defined by lakes and o ceans. smaller than the system scale but sufficiently large such that densities can b e approximated b y smo oth functions. With a p ow er sp ectrum P ( k ) in hand, it is p ossible to calculate the num ber of cities as a function of their p op- ulation N . W e picture cities of area A as discrete ob jects whic h form when the p opulation density as a function of spatial coordinates ρ ( x ), or equiv alently δ ( x ), av er- aged ov er an area A surpasses a critical threshold, δ C . In other words, we choose the surface area A such that the total in tegrated population N = R x ∈ A ρ ( x ) d 2 x = ρ C × A , where the critical densit y ρ C = ¯ ρ (1 + δ c ). This is shown pictorially in Figure 2. [This assump- tion can b e relaxed, allowing for the av erage p opulation densit y of a city to v ary systematically with size. In our mo del, this corresponds to a critical threshold that v aries with A . In this case, the excursion set formalism can b e used with a moving barrier[17]. W e will ignore this sub- tlet y , since Zipf ’s law is still obtained in the limit that δ C  σ . Also, since the numerical v alue of the threshold is not fixed, we could also consider the case where the threshold v aries by coun try . Again, Zipf ’s law would b e obtained for each country .] The counting of cities is now a well-posed question. Computationally , one could find the num b er distribution of cities with the follo wing algorithm. Generate via a Mon te Carlo pro cedure man y realizations of the random field with mean 0 and p ow er sp ectrum P ( k ) = P 0 k − 2 . Find the regions where the random field exceeds a certain threshold. Measure the size of eac h region, and multiply the area of each region by the p opulation densit y thresh- old; define this to b e the p opulation of e ac h city . Rep eat for many Monte Carlo iterations, and then make a his- togram of the size distributions of each region. One can v erify numerically that the resulting n umber distribution n ( N ) would scale approximately lik e n ( N ) ∝ N − 2 , (4) where N is the p opulation of the city and n ( N ) is the n umber density of cities of size N . Ho wev er, we can in fact show analytic al ly that the num ber distribution takes this form using the Press-Schec hter (PS) formalism[10], traditionally used in the context of cosmology to predict the abundance of gravitationally-bound ob jects giv en a p o wer sp ectrum of the fluctuations in the cosmic mat- ter density . Ho wev er, w e emphasize that the formalism is in essence a purely statistical one, which do es not re- quire or emplo y any facts from cosmology . The excur- sion set formalism[18] provides a more rigorous deriv a- tion, but the PS formalism has the b enefit of simplicity . The end result is identical in either case. W e provide a self-con tained pro of of equation (4) in App endix A. By integrating equation (4) with resp ect to N , we find that the num b er of cities ab ov e a certain p opulation threshold scales inv ersely with the p opulation threshold. This statement is equiv alent to Zipf ’s law: the rank of a cit y is inv ersely prop ortional to its size. I I I. CONCLUSION In summary , we hav e presented a deriv ation of Zipf ’s la w and successfully predicted the p ow er sp ectrum P ( k ) of p opulation density fluctuations in the continen tal US. These deriv ations stemmed from tw o fundamental ingre- dien ts: scale-in v ariance and 2D geometry . Remark ably , there is a wide range of p ossible mo dels and an even wider range of initial conditions to which our results are insensitiv e. One such mo del in v olves random walks of the sizes of clusters of p eople on all scales, which can b e view ed as a v ast generalization of Gibrat’s la w. How- ev er, as we hav e emphasized, even this generalization is still a relatively sp ecific example in the class of all models whic h will lead to Zipf ’s law.This sho ws that the origin of these la ws is fundamentally the scale-free nature of clustering in human p opulations. This is an app ealing feature, enabling us to forgo an y fine-tuning arguments in explaining the empirical data. A CKNOWLEDGMENTS The authors would like to thank the anon ymous ref- erees for useful discussions. This work was supp orted in part by NSF grant AST-1312034. App endix A: The Press-Schec hter formalism The Press-Schec h ter formalism (for a p edagogical o verview of the PS formalism and its generalizations, see 4 FIG. 2. (Color online). Schematic illustration of our approach. On the left, a simulated p opulation density map with a p ow er sp ectrum P ( k ) ∝ k − 2 is display ed. Darker pixels indicate higher p opulation densities. On the right, we select and color pink all pixels abov e a certain p opulation densit y threshold from the sim ulated map on the left. In our formalism, a city is identified with each pink cluster, appropriately smo othed on the length scale of the cluster. Scale inv ariance implies that cities of all sizes app ear on the map, as confirmed by visual insp ection. In our formalism, the statistical size distribution of pink cluster gives us the p opulation distribution of cities, which agrees with Zipf ’s law. section 3.4 of [19].) allows us to answer the w ell-p osed question: given a random field with an asso ciated p o wer sp ectrum P ( k ), ho w often do es it exceed the threshold? More sp ecifically , supp ose there is a class of ob jects (e.g. cities) that form when the p opulation density exceeds a certain threshold ρ ( x ) > ρ threshold . F urthermore, let the size R of each ob ject b e defined as the maxim um radius R suc h that the av erage p opulation density ρ circle,R within a circle of radius R centered on the ob ject is given by ρ circle,R = ρ threshold . (A1) It is conv entional to define a smo othed density field δ A ( x ) = Z d 2 k W A ( k ) δ k e − i kx / (2 π ) 2 , (A2) where the low-pass window function W A ( k ) = 1 if k ≤ 1 / √ A and W A ( k ) = 0 otherwise. This smo othed field is simply the original field δ ( x ) with the high-frequency fluctuations subtracted out, leaving b ehind the slo wly- v arying comp onents. F or a fixed x , δ A ( x ) is a random v ariable with prob- abilit y distribution p A ( δ A ). The key insight of the PS formalism is to identify the fraction f A of people living in cities of area A or larger with the cumulativ e probability f = 2 R ∞ δ C p A dδ . This is illustrated in Figure 2. T o make further progress, we m ust assume something ab out the functional form of p A . The conv entional PS formalism assumes that p A is a Gaussian with mean 0 and v ari- ance σ 2 ( A ) ≡ R 1 / √ A k min dk k P ( k ) / (2 π ) ∝ ln k/k min . If each F ourier mo de is statistically indep endent of ev ery other F ourier mo de, the densit y field will b e the sum of man y indep enden t F ourier mo des and will therefore b e approx- imately Gaussian. How ever, for the sake of generalit y , w e will not assume that δ is normally distributed. In- stead, w e only assume that p A has a universal shap e for all A . Since the mean of δ A is zero for all A by defini- tion, and since for any random v ariable θ the asso ciated standard deviation ob eys σ aθ = aσ θ , this allows us to write p A ( δ ) = g ( δ /σ ( A )) /σ ( A ) for some general proba- bilit y density function g . Differentiating f A yields n ( N ), the n umber of cities on Earth’s surface with p opulation N = ¯ ρA p er unit area p er unit p opulation: n ( N ) = − ν g ( ν ) ρ N d ln σ dN ∝ 1 N 2 g ( ν ) ln( N max / N ) , (A3) where w e ha ve defined ν ≡ δ C /σ ( N ), the n umber of stan- dard deviations asso ciated with city formation. Note that for ν  1, g ( ν ) is a slowly v arying function of N for tw o reasons: the first deriv ative of g around ν = 0 is small for small deviations from the mo de, and ν is only a weak function of N . Thus, equation (3) implies that the loga- rithmic slop e d log n/d log N tends to − 2 in the limit of N  N max . This limit is empirically justified, since even the largest cities in the world contain only ∼ 10 − 3 of the w orld’s p opulation. Hence we arrive at equation (4). 5 Although our results are largely independent of the exact form of p A ( δ ), let us briefly comment on its p ossible form. If p A ( δ ) deviates from a Gaussian, this implies that differen t F ourier mo des in human p opulation density are correlated, a generic result of non-linear interactions. Note, ho wev er, that δ ≥ − 1 is strictly b ounded from b elo w, since human p opulation densit y is alwa ys positive-definite: ρ ≥ 0. Hence, p A cannot b e exactly Gaussian. A t some lev el, non-linear in teractions m ust come into play . If the p opulation densit y fluctuations w ere typically small δ . 1, one migh t exp ect that a Gaussian distribution could b e a go o d appro ximation; how ever, everyda y exp erience tells us that p opulation density fluctuations can b e quite large. Indeed for New Y ork City , δ ∼ 300. Hence, a theory of human p opulation density growth must necessarily b e a non-linear. App endix B: Deriv ation of the inv erse-rank friendship law As a second application of our formalism, let us derive the a verage num b er of friends a p erson has in a given region. W e again adopt a simple mo del, where w e de- fine a region to b e a c ommunity if the p opulation density exceeds some critical v alue δ ≥ δ c . This defines geo- graphic equiv alence classes on the inhabited regions, suc h that every p erson is a member of a communit y . Since real-w orld so cial netw orks are highly clustered and only a small fraction of p eople serve as connections b etw een comm unities of friends [20], this assumption should b e a go o d approximation for our purp oses, since the more complicated top ology of real-world friendship netw orks will mainly affect higher order quan tities that inv olv e friends-of-friends and friends-of-friends-of-friends. F ur- thermore, we assume that the av erage num b er of friends D a given p erson has is asymptotically indep endent of the size of the communit y . This second assumption is essen tially the assertion of the existence of the famous Dun bar’s num b er [21, 22], an upp er limit on the n umber of p eople a given p erson can sustain so cial relationships with. T o compute the probability in the mo del, we consider t wo p eople A and B with N AB p eople closer to A than B . If A is a mem b er of a comm unit y with size N c  N AB , A and B are almost certainly friends. On the other hand, if A is in a communit y of size N c  N AB , it will b e nearly imp ossible for A and B to b e friends. There is thus a turno ver scale at ∼ N AB whic h dictates whether or not A and B will b e friends; the probability is therefore deter- mined by tw o indep enden t even ts: the probabilit y that A is in a communit y of size N greater than the turnov er scale and the probability p f = D / N that A and B are friends given that A and B are in a communit y of size N , for large N  D . Since we know from the previ- ous discussions that in such a mo del, the num b er density of comm unities scales asymptotically with ∝ 1 / N 2 , and eac h communit y has N p eople, the probability p c that a randomly c hosen individual is in a communit y of size N scales ∼ 1 / N . Hence, p ( N AB ) = Z g ( N , N AB ) p c ( N ) p f ( N ) dN ∝ Z N >N AB 1 N D N dN ∝ 1 N AB (B1) where g has the prop erties that 0 ≤ g < 1, g ≈ 1 for N  N AB , and g  0 for N  N AB . The details of the function will depend on the geometry of the comm unities but do not concern us here as we are only interested in the scaling. W e ha ve thus deriv ed the in verse-rank friendship law, previously prop osed[23] to fit empirical data. W e stress that our deriv ation is based en tirely on theoretical considerations and therefore provides an explanation for the “ph ysical” origin of the la w. App endix C: Tw o-p oint correlation function In this app endix, w e analytically compute the tw o- p oin t correlation function[24] ξ ( x − y ) = h δ ( x ) δ ( y ) i , whic h is the inv erse F ourier transform of the p ow er sp ec- trum. The correlation function will play an imp ortant role in App endix D. Physically , the correlation function measures the degree to which the existence of an ov er- densit y or under-density at some p osition x increases the lik eliho o d that an ov er-density or under-density will b e found at y . Assuming circular symmetry , the inv erse F ourier transform is a Hankel transform of order 0: ξ ( r ) = Z ∞ 0 k dk 2 π P ( k ) J 0 ( k r ) , (C1) where J 0 is the first Bessel function. T aking P ( k ) = P 0 k − 2 and a long-wa v elength cutoff k m giv es us an inte- gral that can b e written in terms of sp ecial functions ξ = P 0 2 π Z ∞ k m dk J 0 ( k r ) k = P 0 4 π G 23 01  ( k m r / 2) 2  , (C2) where G is the Meijer G function. Defining a reduced area a = k 2 m r 2 / 2 and consider separations that are small compared to the system size (corresp onding to the scale of continen ts) a  1, we can expand ξ ( r ) ≈ P 0 4 π  − γ + 1 2  − ln a + a − a 2 8 + a 3 108  (C3) where γ ≈ 5 . 7721 is Euler’s constant and we only neglect terms O  ( k m r ) 8  . The second term guarantees that ξ  1 for sufficiently small a and ξ < 0 for a & 0 . 51. Most imp ortan tly , we note that for r muc h smaller than the system size, ξ ( r ) → − P 0 4 π ln r. (C4) 6 Since it is p ossible to inv ert a F ourier transform, any 2D mo del which predicts a correlation function that logarith- mically diverges for small r must hav e a p ow er sp ectrum of the form P ∝ k − 2 for k  k m . App endix D: Effectiv e field theory The basic program of effective field theory is the fol- lo wing: given a statistical physics system in a fixed num- b er of dimensions (in this case D = 2), write down the Hamiltonian H = Z d 2 x H ( δ, ∇ δ, ∇∇ δ, . . . ) , (D1) suc h that H contains all terms whic h are consisten t with the symm etries of the system. By universalit y , the macroscopic prop erties of the system should then b e re- flected in the field theory . F or a p edagogical introduction to this approac h, see [12]. In our case, the symmetries are particularly constraining: we w ant the Hamiltonian to be in v arian t under scaling op erations x → λx in addition to translations and rotations of the Euclidean plane. Un- der a change of scale, the p opulation density transforms lik e a scalar, so δ ( x ) → δ ( λ − 1 x ) while d 2 x → λ 2 d 2 x and ∇ δ → λ − 1 ∇ δ . Hence scale inv ariance requires that each term in H contain exactly t wo deriv atives to cancel the λ 2 from the area element. Rotational symmetry then limits us to only one p ossible term ( ∇ δ ) 2 : H = 1 2 Z d 2 x ( ∇ δ ) 2 , (D2) whic h is simply a free scalar field in tw o dimensions. Adding an y in teraction term to H of the form V ( δ ) is not allo wed, as d 2 x V ( δ ) would not transform correctly un- der a scale. Using standard field theory techniques, one can show that the correlation function ξ ( r ) has the form of (C4); hence for w av e v ectors k  k m (corresp onding to physical scales shorter than the system size), we must ha ve that the p o wer sp ectrum P ∝ k − 2 . This concludes our pro of that a scalar random field in 2 spatial dimen- sions will hav e a p ow er sp ectrum P ∝ k − 2 . Let us make some further comments ab out (D2) that ma y b e helpful to readers unfamiliar with effective field theory tec hniques. In particular, we can use (D2) to con- struct alternate theories that also will yield p ow er sp ectra P ∝ k − 2 . F or example, the generic Langevin equation, sp ecialized to the case where the Hamiltonian is given b y (D2), is just the famous diffusion equation with a noise term[12]: ∂ δ ∂ t = − λ ∇ 2 δ + η , (D3) where δ is the fractional population ov er-density and η is a fluctuating random v ariable. Notice that while this mo del also inv olves a diffusion equation, it is con- ceptually distinct from equation (2). Here we think of p opulation as physically diffusing in Cartesian space; in equation (2), the diffusion is not happening in Carte- sian space but in F ourier space. Since (D3) w as deriv ed from a Langevin equation corresp onding to the Hamilto- nian (D2), its equilibrium prop erties must b e describ ed b y (D2); hence it follows that P ( k ) ∝ k − 2 . Note, how- ev er, that the noisy diffusion mo del lacks parameters di- mensions of length to any p ow er. 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