Resistance of the infinite square lattice

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Numbers

$m$, $n$

$r(m,n)$

1, 0:

1/2

comment: Adjacent vertices have resistance $R_0/2$.

2, 0:

0.7267604552648373138489298930198851037243228340763484100186612475288256189261877192790895778699753124

1, 1:

0.6366197723675813430755350534900574481378385829618257949906693762355871905369061403604552110650123438

comment: Opposite corners of a unit square have resistance $2R_0/\pi$.

3, 0:

0.8605627315890238830935793581193106223459370044580904601119674851729537135571263156745374672198518741

2, 1:

0.7732395447351626861510701069801148962756771659236515899813387524711743810738122807209104221300246876

4, 0:

0.9539872947883442913671833859431431808792335783413512405722782575506523137364233912254137213459095788

3, 1:

0.8807515881514569635791020767571071023900958027073310949284652178061684607829470760968232848317613027

2, 2:

0.8488263631567751241007134046534099308504514439491010599875591683141162540492081871472736147533497918

5, 0:

1.025804658157845836332349906448666250143910671961250902923595446182680298436076020390700532962798935

4, 1:

0.9647908947032537230214021396022979255135433184730317996267750494234876214762456144182084426004937530

3, 2:

0.9244131815783875620503567023267049654252257219745505299937795841570581270246040935736368073766748959

6, 0:

1.084230860846665315295151873523401798091394285217370137256191896399237659456000101298366227967150358

5, 1:

1.032500238498186869333532183164060010802507412143141116932955815390415610275940294519511091269067902

4, 2:

0.9919244573750267678057909125448814079823364807003037460734009069467141011096716958310856729552362286

3, 3:

0.9761503176302913927158204153514214204780191605414662189856930435612336921565894152193646569663522605

7, 0:

1.133520582038060714427745574183158938036043719122167609129151589985084094303569975199303470455337937

6, 1:

1.088799101595377355210256006730891002092811374893031018486010274714593122542177204801730454225232279

5, 2:

1.050606299536270562770120679874384865459764283245250746695442491240901398649262338467404935287746640

4, 3:

1.027887453682195223381284128376137875530812599108381907977606502965409257288574736865092506556029625

8, 0:

1.176163777123252286748213062226301834304722158668843952996081675139011581712408306052110308295792564

7, 1:

1.136843845091162627833808680491466059874029216301228173132166394201043568022935746723368672779204414

6, 2:

1.101621461945494608378531289744636139603314585910384646622726992867675652413832676665675824885506443

5, 3:

1.076379040326374005562628334043961903450898654227173477152686249758800230797604686853347152041175988

4, 4:

1.067095999397088727440896851564286770211996100964584189698645811594889005090433149556572544261354024

9, 0:

1.213748300343088571040939431019178766330010480711744150611056328183341270261273742345714881204932618

8, 1:

1.178693113055929930762083621851434816426417217420732026122059391193810481142394753331711440761449849

7, 2:

1.146362583675282510935149519200379482940844553768982038791444320910686574103601053560729325674797590

6, 3:

1.120717862975048004598598953172889207919838131734274782518010884604521514360290109832838584354249263

5, 4:

1.106304545111982231500509574752435664893179602820786471419685120224368752892291562248052581966678423

10, 0:

1.247351142431416930398076630698230484870521563144060565882937202800921251999898000504655780484097184

9, 1:

1.215739140908842533508734015576091373072399100517036041782603217396716123666394331413046718019920363

8, 2:

1.186025689100462274957578729111879998454518394195819936577386278038470651167840629138320063950882057

7, 3:

1.160959338564010532570679377453535733831516018668495398833497618535556424809795161715552741083597447

6, 4:

1.143911611064312871882556811449423054793623268131045607463132677256053749419927914096778619406717173

5, 5:

1.137831529660153321115956301952070931116200387960342611364275742287732026261200498485512012157466507

11, 0:

1.277737316963519654255197926257224552101107526577288796081047125134112990108696151274283748544106691

10, 1:

1.248959476209529748148084582758259310525484122643604658419822678943115373814811054199312246093674714

9, 2:

1.221555674026821884083828426675492599007684581292063331977474471266597369447097775775448304019624270

8, 3:

1.197491385643814774049253348720213095443127224301502349418566928782788179978268933885391185347656519

7, 4:

1.179265521961896840699715688720661149022254164869222424605968339844229430797020549583251868957686415

6, 5:

1.169358514208324410731403029151706197339221173099898751308866364351095299630109434722971442348254590

12, 0:

1.305470137863885592734852427752314929723482772581521774242935998619940744871807845472123982440372924

11, 1:

1.279063993779388046943931323289176396905212885291786422099157649557794981781539379560177615625978327

10, 2:

1.253683627718471481741596361469538987253802941621535603914592646017029137811412505319368870244702980

9, 3:

1.230774238379511246127404600544460037250017888833861745635315743614173565142743637231057563862991679

8, 4:

1.212206276531275042541350687770976612236456595507832316628068074942952078792696307456634372493154894

7, 5:

1.199984861687988915804275878208709195227420777169516375499174988642355470165662815064041742847276145

6, 6:

1.195706054420842534122823124996621608219640259138690410908882049218239952673646511245553394981558538

13, 0:

1.330976569325298684944582114838330464734463540131764555160418029652093376403817425510803821444726326

12, 1:

1.306583332583362015869814834956852351029180011808516872865139419846778306487358902551704179886329339

11, 2:

1.282975849361140769502627949184369373965079880137735361030621374307173256715291410215410287979802566

10, 3:

1.261243511276393525231844487260034665516963182412739064230452059551230551268449781087304642885710373

9, 4:

1.242806382571014801144692139522099789032296567329142236914769514856078159877158058176086123198975554

8, 5:

1.229261815948373754271741574120932415448148425531462255572967516288712544518337688181808312468301089

7, 6:

1.222053594633360657514243220841537019100059345177482070508897734085384605717183587768135347614862485

14, 0:

1.354588028099639297854028453802173979971338728462053881382506267583657986147630186418957259632517534

13, 1:

1.331924055668834924594723788899416474621516329741741282508114926202387387297915835076067021853007424

12, 2:

1.309875143021339499205751799886501602866508059619018012610349105006990111998172550098448099625958682

11, 3:

1.289280632925364050119232312092260508834795634018601405498386096646878795270041205883646566422570273

10, 4:

1.271235546082227322939144674933879128729236265176957501873513751926840706849601775915145571012576559

9, 5:

1.257009469291045592970868594839083377913475519797917383522180488940346288723590512101506985427179085

8, 6:

1.247846656283185465770471135664960476415240809650582946642446512629196340391401118105050149105594232

7, 7:

1.244676806141425714359402744495856796537935534751138548985087385851746659638023906657896103525021026

15, 0:

1.376566957170214780767428269300245971597657522081813001531116418767056186023265673014434414205693887

14, 1:

1.355404292951521862852051715535059741776616925817318984419245310957741191081718823575295401439824962

13, 2:

1.334732027815157134712446490267423340945804841209364717587656944352936675218768188666464684641149067

12, 3:

1.315209362325698076738117925137361345525967505320455098957978681521072209571271698960213245996553756

11, 4:

1.297693808738223829004338886787276650331171968203476097774492271208039163525151933271658088828214397

10, 5:

1.283198481743277136375703186166105410036513342762472608574341162092014952727647331125533429137405910

9, 6:

1.272771196901516680091337479547195897136943743568592433026643762524579497771218970922600076904033788

8, 7:

1.267300017649490771204562268150176573975811724324795027461277037618108713558864225547656859435179566

16, 0:

1.397124747941157163872925942357468582390844538239902421253683690063227150802805433039198006946176623

15, 1:

1.377277526320031330671379340520670662014223410812647851744137858710669803571313536299791195122040695

14, 2:

1.357827561717581898288075278917977850499389234252832922042222191334249587310015736506366129151734197

13, 3:

1.339301350852872216761235093365797435795805741223866653189941554868119614268968632984977487933895964

12, 4:

1.322380322503216540866252495204885834606760586420334324533237969562300316747904406873780830003790105

11, 5:

1.307878733442087402092726064918081129153895387198011159192831266696136835233060344414059387873920653

10, 6:

1.296670178157748227500073409973378004349446198877004389708839140804735980104336692071421772235947343

9, 7:

1.289558483874087434123936777711361729869612445948865012233108907724039381865547561412421400847414492

8, 8:

1.287118124299264470564438414728527293080458106948593601984465344267452472340484316015259784262688515

17, 0:

1.416434189247647582557876958286875334459340377770044773877950265708336672383358751954819655954753153

16, 1:

1.397748922673383146083199270921376511753190126553875954802834037888757872402298653593768978812129725

15, 2:

1.379389932485005532982838106326000422929429068797583466223355667229123964777970995015665986030514204

14, 3:

1.361783993618643062604964803543427896345706101187064519938630842797196518161604938768038444495448553

13, 4:

1.345480019651990592989411154514977160365744517178582276275499750801273054124229705545193576602432481

12, 5:

1.331138099296953664733142014379928182204158354978823825124981174718816839770964289718058408587959784

11, 6:

1.319484543989894978257720172339014274043737882847272105297510458765676384908477823540987624942102522

10, 7:

1.311226489996304115275532801841196436180589826329680411936861179836673085010002642696565957960247150

9, 8:

1.306936230949038169924314561306878012185104489572392176507653650916796231122104406482862709090197464

18, 0:

1.434638335851901149262060352470145439507247359477751280010890024276332720555613465574197029241301861

17, 1:

1.416986836598766008548260769159943657969634806681262697123613674246893409087508054602941793815767064

16, 2:

1.399606579833578081240231031647423144638057750481690849089900928534241126147567590433144919364534521

15, 3:

1.382848062068830821731666774217930034566045879643162242017161690337335342082987116823361700483747404

14, 4:

1.367158999835287313638882067672006264521583549628396262505197934649081528984448268757448460412416648

13, 5:

1.353079405416586300691274961817219106538828191441731864873621544125590756495597513564567528059627208

12, 6:

1.341213935825924415282314535579526658517149254855217951900233918491239450607294894019825888414501169

11, 7:

1.332175328533819868155766678885071304154460690458854920388137509070593273689219363658643451244040922

10, 8:

1.326501969419560931322354340794974706348839970033997325417359161747323704380906953643777207513585844

9, 9:

1.324566346203239843686528711992648319441507435358112766395681189928369365901478794859992443737101006

19, 0:

1.451856966213074743714505853106997181095587728442924144420045808231974321119142700449094014786401232

18, 1:

1.435131093973441135387929299243354621237030665849018100872782011582509944359976204946437223112026530

17, 2:

1.418633140500592170164037548188168164428978056552111958940888381807969147204398607916741317384158847

16, 3:

1.402654323675331475730849201154147479440633750023192016392525627211113520205602105206403395231335306

15, 4:

1.387563998496343215608014985848144339548414598564808965514134624111907365186770428303338976177691555

14, 5:

1.373807987574172383353137326781475661826468981583129288292526520885949178465188002413222844374094003

13, 6:

1.361891515143228561689409351592495421758940912026392069801478730096323953622008056581795282674022566

12, 7:

1.352341584873620456448986604006668756061759869568383807376965310384140624127729406238462237453919806

11, 8:

1.345648695275459922640827137353405750331755183090083356941213087295883000710667582158557984619894212

10, 9:

1.342196461457441517448742862678418626697910381143833356283708728939942500680853183237122178384004548

20, 0:

1.468191376027734854103720556707605945788628682046704104895942524673085606086371539672455186061039588

19, 1:

1.452299076486331485746121251625118669543237436123620596386675341989239478917292898274861921921631740

18, 2:

1.436600126956765897995274823718210717928003061113437829969949005817574168879490401333748147469405455

17, 3:

1.421339018613258692872383568227095137180216607932056459580089918633167884703028385297130408886928350

16, 4:

1.406823634185658307079115430524141601378214762035858514882949971339709727888825328271976552190130948

15, 5:

1.393425297895596419982395670978499457727814202951818842651228900121502861790820999360569191624471218

14, 6:

1.381568247149219499099996606658177818517649982310570183140057704647621566589885227970306197399860937

13, 7:

1.371704472181184031684051302315058103462136219498048279292001753120844040795254590772481516822100947

12, 8:

1.364272602953553510673813899247018958113293313461414077927488060853885731419148776502898093335036184

11, 9:

1.359644880194905380411373630486558032710426758406067104031867617511729293353395234828913186386913897

10, 10:

1.358072650012059914374714767439493448290867360777156229289926946572347639087631749615805875898417445

Definition

The table lists the dimensionless effective resistance $r(m,n)=R_{\mathrm{eff}}((0,0),(m,n))/R_0$ of the nearest-neighbor graph on $\mathbb Z^2$, whose edges all have resistance $R_0>0$ [1]. The infinite network is defined as the limit of periodic square tori with the two vertices held fixed.

Parameters

$m$

— larger displacement coordinate ($m\geq n$, $m>0$)

$n$

— smaller displacement coordinate ($0\leq n\leq m$)

Formulas

(1)

$r(m,n)=\dfrac{1}{(2\pi)^2}\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\dfrac{1-\cos(mx+ny)}{2-\cos x-\cos y}\,dx\,dy$ [1], equation (25).

(2)

$r(m,n)=\dfrac1\pi\int_0^\pi\dfrac{1-e^{-ms(y)}\cos(ny)}{\sinh s(y)}\,dy$, where $s(y)=2\operatorname{arsinh}(\sin(y/2))$ and $\cosh s(y)=2-\cos y$ [2], equations (27) and (28). The integrand has limiting value $m$ at $y=0$.

(3)

$4r(u,v)=r(u+1,v)+r(u-1,v)+r(u,v+1)+r(u,v-1)$ for $(u,v)\ne(0,0)$, using the symmetry convention to interpret signed coordinates [1], equation (32).

(4)

$r(k,k)=\dfrac2\pi\sum_{j=1}^{k}\dfrac1{2j-1}$ for integers $k\geq1$, as follows from the diagonal recurrence in [1], equation (32).

(5)

$r(1,0)=\tfrac12$, $r(1,1)=\tfrac2\pi$, $r(2,0)=2-\tfrac4\pi$, and $r(2,1)=\tfrac4\pi-\tfrac12$ [1], equations (26), (29), and (32).

Comments

(6)

Edges join vertices differing by $(\pm1,0)$ or $(0,\pm1)$. In units of the common edge resistance $R_0$, the positive combinatorial Laplacian is $Lf(v)=4f(v)-\sum_{w\sim v}f(w)$. For unit resistors, the entries are the numerical resistances in ohms.

(7)

For any nonzero displacement $(u,v)$, use $m=\max(|u|,|v|)$ and $n=\min(|u|,|v|)$. Sign changes and coordinate interchange preserve the resistance, so each displacement orbit is listed once.

(8)

The resistance uses a difference of Green functions. In two dimensions the separate origin Green-function integrals diverge, whereas the numerator in (1) cancels the singularity sufficiently for the integral to converge [1].

(9)

Entries are ordered by increasing graph distance $m+n$, then by increasing $n$. The coincident-vertex value $r(0,0)=0$ is a recurrence seed and is not an entry.

Programs

(P1)

Python

import mpmath as mp
mp.mp.dps = 120

def resistance(m, n):
    # Numerical quadrature; the attached generator provides certified bounds.
    def integrand(y):
        if not y:
            return mp.mpf(m)
        s = 2*mp.asinh(mp.sin(y/2))
        decay = mp.exp(-m*s)
        return (-mp.expm1(-m*s) + 2*decay*mp.sin(n*y/2)**2)/mp.sinh(s)
    return mp.quad(integrand, [0, mp.pi/4, mp.pi/2, mp.pi])/mp.pi

print(mp.nstr(resistance(2, 1), 100))

References

[1]

József Cserti, Application of the lattice Green’s function for calculating the resistance of infinite networks of resistors, American Journal of Physics 68 (2000), 896–906. (arXiv) (doi)

Links

[2]

Cserti: square lattice resistances, version 4, section II.C

Similar tables

Kirchhoff indices of connected graphs — sums effective resistances over vertex pairs in finite connected graphs

Watson integrals of the cubic lattices — holds convergent origin lattice Green-function integrals in three dimensions

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds all 120 nonzero displacement orbits with $m+n\leq20$, normalized by the common edge resistance $R_0$: 20 axial, 10 diagonal, and 90 asymmetric displacements. The graph-distance cutoff 20 is a noncanonical finite reference choice that covers every short-separation direction before extending one direction further)

How they were obtained:

The generator evaluates Cserti's recurrence (32) exactly over $\mathbb Q[t]$, with $t=1/\pi$, then evaluates each linear polynomial using Arb ball arithmetic. An exactly zero coefficient of $1/\pi$ yields an exact rational; this occurs only at $(m,n)=(1,0)$ in the stored range. Working precision starts at 461 bits for 100 requested decimal digits and increases if an exact endpoint-width check detects inadequate relative accuracy. The smallest measured relative accuracy is at least 413 bits, and every entry used 461 working bits. All written decimal intervals contain the computed balls.

Independent mpmath quadrature of (2) at 120 and 145 decimal working digits agrees for all 120 entries within $10^{-112}$ in absolute value. The zero, adjacent, and diagonal seed values serve as controls. Exact coefficients satisfy (3) at all selected displacements and (4) on all ten diagonals. Discrete Fourier solves on periodic square networks of side 128, 256, and 512 provide an additional finite-network check, with the known finite-torus adjacent resistance as a control. Quadrature and finite-network computations are checks; certification comes from the exact recurrence and Arb enclosure of $\pi$.