Arbitrage-free Self-organizing Markets with GARCH Properties: Generating them in the Lab with a Lattice Model
aa r X i v : . [ q -f i n . C P ] D ec Arbitrage-free Self-organizing Markets with GARCHProperties: Generating them in the Lab with a Lattice Model
B. Dupoyet a,b , H.R. Fiebig a,c, ∗ , D.P. Musgrove a,c a Florida International University, Miami, Florida 33199, USA b Department of Finance c Department of Physics
Abstract
We extend our studies of a quantum field model defined on a lattice having the dilationgroup as a local gauge symmetry. The model is relevant in the cross-disciplinary areaof econophysics. A corresponding proposal by Ilinski aimed at gauge modeling in non-equilibrium pricing is realized as a numerical simulation of the one-asset version. Thegauge field background enforces minimal arbitrage, yet allows for statistical fluctuations.The new feature added to the model is an updating prescription for the simulation thatdrives the model market into a self-organized critical state. Taking advantage of someflexibility of the updating prescription, stylized features and dynamical behaviors of real-world markets are reproduced in some detail.
Keywords:
Econophysics, Financial markets, Statistical field theory, Self-organizedcriticality
PACS:
1. Introduction
The analysis and modeling of financial price time series has a long history [1, 2, 3]and has attracted considerable interest at an accelerated pace in the last two decades.Technological advances have made it possible to collect and process vast amounts ofdata. As a result, various stylized facts about the statistics of financial data have beendiscovered [4, 5, 6]. These features are mostly concerned with scaling laws, akin tofindings in many systems described by statistical physics. There, scaling behavior arisesfrom the interaction of many units in such a way that a critical state is reached. Thus, onemay ask if a financial market, for example, can be modeled based on similar principles. Ingeneric terms, the building blocks could be many individual agents with suitable mutualinteractions. Indeed, Lux and Marchesi have shown that scaling laws can arise in such asetting [7].When building a microscopic model it is prudent to rely on a theoretical foundationsupported by evidence. In the present work, we will employ two such principles. First, ar-bitrage opportunities will be annihilated during the time evolution of the market, though ∗ Corresponding author
Preprint submitted to Elsevier November 21, 2018 dmitting statistical fluctuations. Second, the dynamics of the model will drive it into aself-organized critical state, thus naturally giving rise to scaling behavior. Both aspectshave been investigated separately in previous work, see [8] and [9] respectively. Here, wemerge those elements into a microscopic market model, using numerical simulation tostudy its characteristics.The next section gives an overview of the model’s dynamics and definitions.
2. Lattice model
Following a proposal by Ilinski [10] we define a lattice field theory with a laddertopology as shown in Fig. 1. In physics terms there are matter fields Φ( x ) ∈ R + definedon sites x = ( i, j ), where j = 0 . . . n means discretized time, and i = 0 , µ ( x ) ∈ R + which live on links starting at site x in temporal µ = 0 or spatial µ = 1direction. Those are represented by arrowed lines in Fig. 1. (1,j) Φ (0,j) Φ (0,j) Θ (1,j) Θ (0,j) Θ Θ (0,j+1)(0,j) Θ j=nj=1 i=0 i=1j=0 (1,j) Θ (1,j) Φ (1,j+1) Φ (0,j) Θ (0,j) Φ (0,j+1) Φ = A= BC == D ti m e Figure 1: Left: Illustration of the ladder geometry of the lattice model and the label scheme for thefields. Right: Depiction of the gauge invariant elements A, B, C, D used in the action.
As a model for a financial market, again following [10], we interpret the matter field asinstances of an account value, in some unit. At i = 0 it could be a cash account, whereasat i = 1 the value may be interpreted as the number of shares owned in some financialinstrument. The spatial links Θ (0 , j ), connecting cash and shares, are simply conversionfactors between the corresponding units. Temporal links Θ (0 , j ), which connect cashvalue sites one time step apart, mean interest rate factors. Similarly, temporal linksΘ (1 , j ), starting from a shares site, carry information about the change in share valueone time step apart.The rationale behind such a model is to describe a market that dynamically evolvesindependent of the trading units being used. For example, in comparable markets, thedynamics should not depend on the specific, notably arbitrary, currency unit being used2n transactions. This, at least, is the hypothesis which should apply to markets tradingin like instruments.Mathematically, this idea is implemented by a quantum field theory with local gaugeinvariance. Such has been worked out in great detail in the context of financial markets[10]. In a previous work we have studied some aspects of those ideas using numericalsimulation [8]. Since the current work is directly building on the latter, we refer thereader to [8] for the technical details. In particular, we shall use the nomenclaturetherein. However, to keep the presentation self contained, the essential building blocksare discussed in what follows.The dynamics of the model derives from an action S [Θ , Φ , ¯Φ] for the lattice fields thatis invariant with respect to local gauge transformationsΦ( x ) → g ( x )Φ( x ) (1)¯Φ( x ) → ¯Φ( x ) g − ( x ) (2)Θ µ ( x ) → g ( x )Θ µ ( x ) g − ( x + e µ ) , (3)where ¯Φ( x ) = 1 / Φ( x ) and g ( x ) ∈ G is an element of the dilation group G = R + , i.e.multiplication by positive real numbers. Those carry out conversions between (arbitrary)units. The action is constructed from the elements depicted in Fig. 1. These are thesmallest gauge invariant objects that can be assembled from the fields.The diagram associated with label A is known as the elementary plaquette P µν ( x ) = Θ µ ( x )Θ ν ( x + e µ )Θ − µ ( x + e ν )Θ − ν ( x ) . (4)Its value is interpreted as the gain (or loss) realized through an arbitrage move [10].The global minimum of the classical action S [Θ , Φ , ¯Φ] corresponds to zero arbitrage [8].Quantization of the field is done through the usual path integral formalism. The partitionfunction thus is defined as the functional integral Z ( β ) = Z [ D Θ][ D Φ] e − βS [Θ , Φ , ¯Φ] . (5)In this way stochastic fluctuations about zero arbitrage are allowed. Their magnitude isregulated by the parameter β .Diagrams B,C,D are gauge invariant elements of the form R µ ( x ) = ¯Φ( x )Θ µ ( x )Φ( x + e µ ) , (6)where e µ is a unit vector in direction µ . Diagram C, for example, gives the value of theinvestment instrument at time j + 1 divided by its value at j , provided we adopt theabove interpretation of the fields. It is a measure for the relative change of the assetvalue during one time step R (1 , j ) = ¯Φ(1 , j )Θ (1 , j )Φ(1 , j + 1) . (7)We also define the related quantity r j +1 = log R (1 , j ) , (8)commonly called the ‘return’, indicating a gain ( >
0) or a loss ( <
0) at the end of thetime step. 3 . Updating strategy
The generation of lattice field configurations as implemented in [8] follows a standardprocedure. Based on the action S [Θ , Φ , ¯Φ] the field components are updated througha heatbath algorithm [11, 12] linked to the partition function (5). Periodic boundaryconditions (in the time direction) are imposed on all fields as well. However, in contrastto [8], the updating strategy is modified in two respects.First, we do set constraints on the fields that live on the axis i = 0, see Fig. 1. Thereasoning here is that we wish to design the model such that the axis describes a cashaccount subject to accumulating interest. The interest rate is endogenously determined.Even at 10% annually the daily rate factor is 1 . n translates to typically a day, or so, we wish to set a constraint accordingly. Ina gauge model this is not straightforward, because the meaning of the field componentsis gauge dependent. To remedy this situation, gauge fixing is called for. With referenceto (1-3) define a gauge transformation along the axis i = 0, g (0 , j ) = ¯Φ(0 , j ) , (9)with g ( x ) on all other sites being arbitrary. The gauge transformed fields along the axis, i = 0, then are Φ ′ (0 , j ) = g (0 , j )Φ(0 , j ) = 1 (10)¯Φ ′ (0 , j ) = ¯Φ(0 , j ) g − (0 , j ) = 1 (11)Θ ′ (0 , j ) = ¯Φ(0 , j )Θ (0 , j )Φ(0 , j + 1) = R (0 , j ) . (12)In the last equation we recognize the link variable as the (gauge invariant) return (6) ofthe cash holding during one time step. We therefore setΘ ′ (0 , j ) = 1 . (13)In our simulation we choose a random start for the lattice fields. From there, the con-straint R (0 , j ) = 1 is then implemented by applying the gauge transformation (9), andthen setting the axis links to one (13). During the subsequent updating procedure theaxis fields Φ(0 , j ) and Θ (0 , j ) are never changed. Nonetheless, the right-hand side of(13) may differ from one, depending on the interest rate factor desired.The next step is to run a heatbath algorithm with the lattice action S [Θ , Φ , ¯Φ] untilequilibrium is reached [8]. The lattice field configurations then model a market environ-ment where arbitrage opportunities exist only briefly, subject to fluctuations due to thequantum nature of the fields. In economic terms this model describes an efficient market.Equilibrium, however, does not seem to be realized in the real world [13].Second, subscribing to this paradigm, we introduce a new element which is appliedpost equilibrium. In [9] we have studied a simple model where market instances livealong a linear chain in time direction. The sites carry fields r j ∈ R + which are directlyinterpreted as returns, thus having the same meaning as (8). There is no gauge field inthe simple model. The key ingredient is an updating algorithm that mimics the popularBak Sneppen evolutionary model [14, 15, 16]. The quantity v j = r j ( r j +1 − r j − ) (14)4urned out to be essential to the field dynamics. In the context of [9] the updatingstrategy consists in finding the site j s for which the absolute value of (14) is maximal | v j s | = max {| v j | : j = 0 . . . n } , and then replace r j s and its neighbors r j s ± with randomnumbers. Iterating this process leads to a self-organized critical state and produces pricetimes series, and related statistics, which are almost indistinguishable from those in realmarkets [8].In view of those results it seems desirable to replicate this updating strategy withinthe framework of the gauge model as closely as possible. Towards this end, we still dodefine the ‘fitness’ measure v j as in (14), however, the returns are now given by (8), and(7). Their composition is illustrated in Fig. 2. The updating prescription proceeds withfinding the ‘signal’ V = max {| v j | : j = 1 . . . n } (15)of the field configuration, and the site j s of its location | v j s | = V . (16)We then update the three field components Φ(1 , j s − , Θ (1 , j s − , Φ(1 , j s ), whichenter the return r j s , and the two next-neighbor links Θ (1 , j s ) , Θ (1 , j s − , j s + 1) and Φ(1 , j s −
2) are left unchanged. In this way only the threereturns r j s − , r j s , r j s +1 are affected. This strategy most closely resembles the updatingprescription used in [8]. j+1 r j r j−1 r (1,j−2) Θ (1,j) Θ Θ (1,j−1) (1,j) Φ (1,j−1) Φ Figure 2: Left: Illustration of the returns involved in the ‘fitness’ criterion (14) (left), and the fieldcomponents subject to updating (right), done at the ‘signal’ site j = j s . Updating those field components consists in drawing random numbers from certainprobability distributions. We have chosen those based on the lattice action S [Θ , Φ , ¯Φ]mentioned above. Heatbath steps using the corresponding Boltzmann-like distribution,see (5), are applied to the various field components. Essentially, the probability distri-bution for a given field component is given by its local environment. It is convenient torewrite the fields as Θ µ ( x ) = e θ µ ( x ) and Φ( x ) = e φ ( x ) . (17)Then, after a gauge transformation, the probability densities for the gauge fields and thematter fields, respectively, have the form p Θ ( θ µ ( x )) ∝ exp( − β p L Θ ¯ L Θ cosh( θ µ ( x )) ) (18) p Φ ( φ ( x )) ∝ exp( − β p L Φ ¯ L Φ cosh( φ ( x )) ) . (19)5hese results are derived in detail in Appendix Appendix A. The coefficients L Θ , ¯ L Θ and L Φ , ¯ L Φ are independent of Θ µ ( x ) and Φ( x ), respectively. The products L Θ ¯ L Θ and L Φ ¯ L Φ are gauge invariant and, together with the parameter β , determine the varianceof the probability distributions for the field components. Those distributions stronglydepend on the local environment at the location of the fields.Now, at each updating step we randomly draw fields φ ′ (1 , j ), j s − ≤ j ≤ j s , from(19). Relevant averages considered are a θ = 13 j s X j = j s − θ (1 , j ) and a φ = 12 j s X j = j s − φ ′ (1 , j ) . (20)Updating the fields then is accomplished by replacing θ (1 , j ) ←− θ (1 , j ) − χa θ , j s − ≤ j ≤ j s (21) φ (1 , j ) ←− φ ′ (1 , j ) − a φ , j s − ≤ j ≤ j s . (22)By including the parameter χ we have introduced a novel feature to the updating process.While χ = 1 essentially mirrors the strategy in [9], deviations from that value introducevery interesting features to the model. We will be able to describe a range of differentreturns distributions and time series, as will be described in the next section.Finally, we apply heatbath updates to the two spatial (horizontal) link variables θ (0 , j ), j s − ≤ j ≤ j s , which connect to the affected matter fields, see (22). Theselinks occur in three elementary plaquettes P (0 , j −
1) = Θ (0 , j − (1 , j − − (0 , j )Θ − (0 , j − , (23)see (4), where j s − ≤ j ≤ j s + 1. The reason is that updating θ (1 , j ), as prescribed by(21), changes the plaquettes (23) and thus upsets the no-arbitrage environment of thelattice fields. Updating the above links with the lattice action S [Θ , Φ , ¯Φ] rectifies thiscircumstance.
4. Results
The simulations were done on a lattice of size n = 782 with gauge field couplingparameter β = 1, and the matter field couplings d ± µ = ¯ d ± µ = 1. These parameters are thesame as in [8], with the one asset model m = 1. The number of heatbath update stepswas 10 to equilibrate the field from a random start. Final configurations were reachedafter 4 × ‘signal’ updates.First, we discuss the effect of the parameter χ in (21). A suitable observable (orderparameter) is the gauge invariant link along the asset axis (7). Using the notation (17)we have R (1 , j −
1) = exp ( − φ (1 , j −
1) + θ (1 , j −
1) + φ (1 , j )) = exp( r j ) . (24)The updating algorithm, described in Sect. 3, employs symmetric probability distributionfunctions for θ and φ . Consequently, the probabilities for realizing a gain r j > r j < L j = 12 (exp( r j ) + exp( − r j )) − r j ) − L = 1 n n X j =1 L j . (26)Numerically, the value of L is particular to a distinct lattice field configuration. We denotethe (stochastic) average over field configurations with angle brackets, here h L i . In Fig. 3the dependence of h L i on the parameter χ is displayed. The plot symbols ‘ • ’ indicatedata points from simulations at χ = 10 k , k = − , − . . . + 1. The errors come from 48field configurations. The line curve in Fig. 3 is a four parameter fit, a . . . a , to thoseeight data points with y = a tanh[ a ( − log ( x ) + a )] + a . Remarkably, there clearlyis a transition region in the, approximate, range 10 − < χ < − . For small values χ ≪ − the average link operator saturates at 0 . − ≪ χ it tends to 0 . r ) − L forsimplicity, see (25), this corresponds to returns r ≈ ± .
68 and r ≈ ± .
14, respectively,for the above limits of χ . Those limits may describe valid markets, which could be seen asvolatile and calm, respectively. In view of this observation the transition region becomesparticularly interesting. It opens up the possibility to simulate markets with a wide rangeof features between those extremes. Below, we will present results for χ = 0 . , . Figure 3: Expectation value h L i of the gauge invariant average link operator (26) as a function of theupdate parameter χ in (21). The plotting symbols ‘ • ’ and ‘+’ correspond to specific values of χ forwhich simulations were made. The symbol ‘ ◦ ’ indicates the symmetry point χ = 0 . During a simulation, the evolution of the lattice towards a critical state can be mon-itored, for example, by observing the signal V , see (15), as a function of the updatingstep counter, say s = 0 , , . . . . Writing V ( s ) we follow [16] and define the ‘gap’ function G ( x ) = min { V ( s ) : s ∈ N ∪ { } and s ≤ x } with x ∈ R + ∪ { } . (27)This is a decreasing piecewise constant function with discontinuities at certain discretevalues x k , k ∈ N . The set of update steps between x k − and x k is called an avalancheof length Λ k = x k − x k − . Eventually, as x → ∞ , the avalanche size diverges and the7ystem has reached criticality [16]. For an elaboration on these concepts, presented in acontext close to the current work, we refer the reader to [9]. We here only show a keyresult.In Fig. 4 the frequency distribution ∆ N/ ∆Λ of the avalanche sizes is displayed. Here∆Λ is a binning interval for the avalanche sizes and ∆ N is the count of avalanches withinthat interval. We have used 10000 bins with ∆Λ = 1. The data points and errors comefrom an ensemble average over 2400 independent lattice simulations with 4 × updatesteps each. The log-log plot clearly shows power law behavior. A power law indicatesscaling, which is a signature feature of a critical system. Figure 4: Frequency distributions of avalanche sizes for two values χ = 10 − , − of the updateparameter. For the time being, we continue to present results for the two update parameters χ = 10 − and χ = 10 − . These values correspond to the boundaries of the transitionregion, see Fig. 3. Inside that region, the frequency distributions of avalanche sizes arealmost indistinguishable from the results shown in Fig. 4. Examples of model marketsfor suitable update parameters in the transition region are discussed below.The gains distributions produced by the lattice model with χ = 10 − , − are shownin Fig. 5. The gauge invariant returns r , as defined in (8), are put into bins of size∆ r , and ∆ c/ ∆ r is the number of counts per bin. The errors are obtained from 2400independent simulations. While both histograms possess fat tails, we observe a distinctdifference of the qualitative features for the distributions. At χ = 10 − a distinctlypointed central peak sits on very broad bulging tails. Looking at χ = 10 − the centralpeak has broadened such that its very top is almost Gaussian while the tails look linear.This means that the two distributions cannot be mapped into each other by simplescale transformations applied to the axes. The two gains distributions describe genuinelydifferent markets. Interestingly, this matches our previous assessment of the the regions χ ≪ − and 10 − ≪ χ discussed in the context of Fig. 3.Examples of returns time series for each of the two parameters χ are displayed inFig. 6. By visual inspection, both of those exhibit volatility clustering, yet displaydifferent dynamical behavior. In the realm of χ = 10 − the time series appears to favorone side of the zero mark for short periods of time, as compared to the series of χ = 10 − which smoothly fluctuates about zero in either direction.For a closer investigation, we have selected the parameters χ = 0 . , . igure 5: Lattice generated gains distributions of the returns r for the values χ = 10 − , − of theupdate parameter.Figure 6: Examples of returns time series r j versus the lattice time j at χ = 10 − and χ = 10 − . choice of these values reflects our observation that model market characteristics, say thegains distribution for example, hardly change as χ is decreased from ≈ . ≈ . h L i curve in Fig. 3as χ further decreases below ≈ . χ = 0 . χ = 0 . × each. Again, thedistributions clearly exhibit fat tails but are otherwise different in their shapes. The χ = 0 . χ = 0 . χ = 0 . χ = 0 . × updates each. Compared to the χ = 0 . χ = 0 . igure 7: Gains distributions from simulations with update parameter χ = 0 . χ = 0 . as well as bigger swings between them.We now turn to gauging the ability of our model to replicate various features of fi-nancial markets. One crucial aspect of financial markets returns is their volatility, aswell as how this volatility evolves over time. Volatility is more relevant today than ever,with large spikes possibly occurring in short periods of time. Financial markets returnsgenerally display volatility ‘clusters’. These clusters indicate that once the volatility ishigh, it tends to remain high for a while, and that similarly, once it has come down,it tends to remain low for some time. A convenient and well-accepted way of modelingsuch characteristics is through the use of the Auto Regressive Conditional Heteroskedas-ticity (ARCH) model pioneered by Engle [2] or through the use of the more encompass-ing Generalized Regressive Conditional Heteroskedasticity (GARCH) model proposed byBollerslev [3].Whether working on pricing a derivative product, attempting to hedge an exposure,optimizing a portfolio in a mean-variance framework, or estimating the Value-At-Risk ofa position, the ability to capture and model the stochasticity and the clustering prop-erties of the volatility is paramount. Not doing so can lead to the wrong probabilitydistribution being used, since volatility clusters impact the shape of returns distributionsin two important ways. First, the fact that a period of calm statistically tends to befollowed by another period of calm indicates that there will be a fairly large amount ofprobability mass around the mean (return). Graphically this phenomenon translates intoa probability distribution function that is higher than the Gaussian one in the vicinityof the mean. Second, the fact that a period of extreme movements statistically tends tobe followed by another period of extreme movements indicates that there will be non-negligible amounts of probability mass in the tails areas. Graphically this phenomenontranslates into a probability distribution function that is higher than the Gaussian onein the vicinity of the tails.The fairly wide-ranging GARCH specification models the volatility as both a functionof past squared return shocks and of past levels of itself. If both past return shocks andpast volatility levels are low, for instance, the odds are that the next volatility levels willremain low. If past volatility levels are low but recent past return shocks are high, thelevels of volatility will likely increase. If past volatility levels are high but recent pastreturn shocks are low, the levels of volatility will perhaps decrease. Finally, if both past10 igure 8: Samples of returns time series for the update parameter χ = 0005. The corresponding gainsdistribution is shown in Fig. 7. igure 9: Samples of returns time series for the update parameter χ = 0013. The corresponding gainsdistribution is shown in Fig. 7. p and q allowed for past shocks and past volatility levels are limitless. However, Hansen andLunde [17] explore the ability of 330 different ARCH/GARCH models to capture thefeatures of various financial returns and come to the conclusion that a GARCH(1,1)model performs just as well as the more ‘sophisticated’ ones. Our specification for thevolatility σ t at time t is thus a GARCH(1,1) described by the following equation: σ t = α + α ǫ t − + β σ t − , (28)where ǫ t − and σ t − are the one-period lagged squared return shock and one-periodlagged variance, respectively.We simulate 100 lattice time series for two different levels of our tuning parameter χ . In the first set of simulations, χ is set equal to a value of 0 . χ is set equal to a value of 0 . χ value of 0 . χ value of 0 . χ = 0 . χ = 0 . β parameter is between 0 .
85 and 1 . β parameteris between 0 .
85 and 1 . α parameter is below 0 .
15 in about 99% of thecases, a result also consistent with Nasdaq GARCH(1,1)-estimated figures in [9]. Finally,in Fig. 11 the α parameter is below 0 .
15 in about 99% of the cases, although a higherproportion is above 0 .
05 when compared to the values it takes in Fig. 10. These resultsindicate that the lagged return shock feedback is also strongly consistent across variousmarket conditions and various simulations. The α parameter is consistently very low inboth cases, as it should at these high frequencies.It is also interesting to note that for each parameter, we obtain some ‘outliers’ inabout 5% to 20% of the cases, although they are outliers only in the sense that they differfrom the other 95% to 80% of the estimates that are themselves incredibly consistent,and are not outliers in the sense that their values would be considered too extreme orunreasonable. It is important to note that when running multiple simulations, one isbound to obtain some results that differ somewhat from the estimates’ consensus. Forinstance, even if one was to simulate an exact GARCH(1,1) process many times andsubsequently estimate the parameters from the simulated data, some percentage of the13 igure 10: GARCH(1,1) fit parameters for a sample of 100 lattice time series at χ = 0 . c has been subject to sorting with respect to β . igure 11: GARCH(1,1) fit parameters for a sample of 100 lattice time series at χ = 0 . c has been subject to sorting with respect to β .
5. Conclusion
The subject of this work has been to explore a class of models designed to simulatethe properties of financial markets. The output of the model is a time series of returns,from which gains distributions and related features could be derived. The potential ofthe model for replicating market dynamics, as described by standard financial analysistools, was the primary aim of this study.The production of our time series has been done by numerical simulation based on alattice description of fields in time-asset space. We have restricted ourselves to a one-assetmodel linked to an interest rate account. The dynamics are based on a gauge invariantlattice action which, when quantized, gives rise to eliminating arbitrage opportunities upto stochastic fluctuations, thus reflecting real market conditions. The second pillar ofthe model is an updating prescription that evolves the lattice fields into a self-organizingcritical state. This appears to be an essential element for reproducing certain stylizedfeatures of real markets.As a third feature, a parameter has been introduced as a tuning tool, through whicha variety of market characteristics, ranging from quiescent to volatile markets, can bemodeled.An extensive analysis of a very large number of time series features evaluated bya GARCH(1,1) analysis was performed. It turns out that close to 100% of the latticemodel-generated time series give rise to sensible analysis parameters, rendering the modelresults almost indistinguishable from historical market data. In particular we could verifythis observation across various market conditions and varying returns distributions.We conclude that the model shows promise as a modeling tool for financial time seriesand look forward to further development and applications.
Appendix A. Gauge fixing
In the notation of [8], the probability density for a particular gauge field componenthas the form p Θ ( θ µ ( x )) ∝ exp( − β ( ¯ L Θ exp( θ µ ( x )) + exp( − θ µ ( x )) L Θ ) ) (A.1)where L Θ and ¯ L Θ are positive coefficients independent of θ µ ( x ). They reflect the (local)environment of the link variable. Under a gauge transformation, writing g ( x ) = e h ( x ) , (A.2)we have θ µ ( x ) → θ ′ µ ( x ) = h ( x ) + θ µ ( x ) − h ( x + e µ ) (A.3) L Θ → L ′ Θ = e h ( x ) L Θ e − h ( x + e µ ) (A.4)¯ L Θ → ¯ L ′ Θ = e h ( x + e µ ) ¯ L Θ e − h ( x ) . (A.5)16he transformation laws for L Θ and ¯ L Θ can be derived directly by an examination ofthe lattice action S [Θ , Φ , ¯Φ]. They are also obvious from the fact that the action is gaugeinvariant and the arguments of the exponential functions of (A.1) are made up frominvariant contributions to it. We now choose the gauge transformation by requiring thenew coefficients to be equal L ′ Θ ¯ L ′ Θ = e h ( x ) L Θ ¯ L Θ e − h ( x + e µ ) = 1 , (A.6)or h ( x + e µ ) − h ( x ) = 12 log( L Θ ¯ L Θ ) . (A.7)A solution of (A.7) is h ( x + e µ ) = λ + 14 log( L Θ ¯ L Θ ) (A.8) h ( x ) = λ −
14 log( L Θ ¯ L Θ ) , (A.9)where λ is a real parameter . On all other sites, besides x and x + e µ , the gaugetransformation function h ( x ) is arbitrary. (For definiteness one may choose it to bezero.) In the new gauge, using (A.6), we have¯ L ′ Θ e θ ′ µ ( x ) + e − θ ′ µ ( x ) L ′ Θ = 2 q L ′ Θ ¯ L ′ Θ cosh( θ ′ µ ( x )) . (A.10)Thus, the probability distribution function is p ′ Θ ( θ ′ µ ( x )) ∝ exp( − β q L ′ Θ ¯ L ′ Θ cosh( θ ′ µ ( x )) ) . (A.11)Dropping the primes gives (18). Citing L Θ ¯ L Θ = L ′ Θ ¯ L ′ Θ , we note that the variance of thethe probability distribution function is not altered by the gauge transformation.Again, in the notation of [8], the probability density for a particular matter fieldcomponent has the form p Φ ( φ µ ( x )) ∝ exp( − β ( ¯ L Φ exp( φ µ ( x )) + exp( − φ µ ( x )) L Φ ) ) (A.12)where L Φ and ¯ L Φ are positive coefficients independent of φ µ ( x ), reflecting the (local)environment of the field variable. In this case, changing the gauge (A.2) entails thetransformations φ ( x ) → φ ′ ( x ) = h ( x ) + φ ( x ) (A.13) L Φ → L ′ Φ = e h ( x ) L Φ (A.14)¯ L Φ → ¯ L ′ Φ = ¯ L Φ e − h ( x ) . (A.15) It may be used to control the effect of the gauge transformation on the matter fields φ ( x ) and φ ( x + e µ ). L ′ Φ / ¯ L ′ Φ = 1 leads to h ( x ) = −
12 log( L Φ ¯ L Φ ) , (A.16)while the gauge function is arbitrary on all sites other than x . Proceeding in the mannerabove we have ¯ L ′ Φ e φ ′ µ ( x ) + e − φ ′ µ ( x ) L ′ Φ = 2 q L ′ Φ ¯ L ′ Φ cosh( φ ′ µ ( x )) , (A.17)and thus obtain the probability distribution function for the matter field p ′ Φ ( φ ′ µ ( x )) ∝ exp( − β q L ′ Φ ¯ L ′ Φ cosh( φ ′ µ ( x )) ) . (A.18)Dropping the primes gives (19). Again, because of L Φ ¯ L Φ = L ′ Φ ¯ L ′ Φ , the gauge transfor-mation does not change the variance of the distribution. References [1] Louis Bachelier. Th´eorie de la sp´eculation.
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