{
  "Title": "Resistance of the infinite square lattice",
  "Definition": "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 $R_0>0$ CITE{Cserti}. The infinite network is defined as the limit of periodic square tori with the two vertices held fixed.",
  "Parameters": {
    "m": {
      "type": "Z",
      "display": "$m$",
      "title": "larger displacement coordinate",
      "constraints": "$m\\geq n$, $m>0$"
    },
    "n": {
      "type": "Z",
      "display": "$n$",
      "title": "smaller displacement coordinate",
      "constraints": "$0\\leq n\\leq m$"
    }
  },
  "Tags": [
    "physics",
    "lattice sums",
    "random walks"
  ],
  "References": {
    "Cserti": {
      "bib": "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": "cond-mat/9909120",
      "doi": "10.1119/1.1285881"
    }
  },
  "Links": {
    "CsertiPDF": {
      "title": "Cserti: square lattice resistances, version 4, section II.C",
      "url": "https://arxiv.org/pdf/cond-mat/9909120v4"
    }
  },
  "Data properties": {
    "type": "R",
    "rigour": "proven",
    "digits": "100",
    "complete": "no",
    "complete-note": "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",
    "rigour details": "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.\n\nIndependent mpmath quadrature of CITE{formula-integral} 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 CITE{formula-harmonic} at all selected displacements and CITE{formula-diagonal} 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 $\\pi$."
  },
  "Display properties": {
    "number-header": "$r(m,n)$",
    "group parameters": [
      [
        "m",
        "n"
      ]
    ]
  },
  "Numbers": [
    {
      "comment": "Adjacent vertices have resistance $R_0/2$.",
      "params": {
        "m": "1",
        "n": "0"
      },
      "number": "1/2"
    },
    {
      "params": {
        "m": "2",
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    },
    {
      "comment": "Opposite corners of a unit square have resistance $2R_0/\\pi$.",
      "params": {
        "m": "1",
        "n": "1"
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    {
      "params": {
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    {
      "params": {
        "m": "10",
        "n": "10"
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      "number": "1.358072650012059914374714767439493448290867360777156229289926946572347639087631749615805875898417445"
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  ],
  "Keywords": [
    "effective resistance",
    "resistance distance",
    "infinite grid of resistors",
    "square resistor lattice",
    "lattice Green function"
  ],
  "Comments": {
    "comment-network": "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.",
    "comment-symmetry": "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.",
    "comment-green": "The resistance uses a difference of Green functions. In two dimensions the separate origin Green-function integrals diverge, whereas the numerator in CITE{formula-fourier} cancels the singularity sufficiently for the integral to converge CITE{Cserti}.",
    "comment-order": "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."
  },
  "Formulas": {
    "formula-fourier": "$r(m,n)=\\dfrac{1}{(2\\pi)^2}\\int_{-\\pi}^{\\pi}\\int_{-\\pi}^{\\pi}\\dfrac{1-\\cos(mx+ny)}{2-\\cos x-\\cos y}\\,dx\\,dy$ CITE{Cserti}, equation (25).",
    "formula-integral": "$r(m,n)=\\dfrac1\\pi\\int_0^\\pi\\dfrac{1-e^{-ms(y)}\\cos(ny)}{\\sinh s(y)}\\,dy$, where $s(y)=2\\operatorname{arsinh}(\\sin(y/2))$ and $\\cosh s(y)=2-\\cos y$ CITE{CsertiPDF}, equations (27) and (28). The integrand has limiting value $m$ at $y=0$.",
    "formula-harmonic": "$4r(u,v)=r(u+1,v)+r(u-1,v)+r(u,v+1)+r(u,v-1)$ for $(u,v)\\ne(0,0)$, using the symmetry convention to interpret signed coordinates CITE{Cserti}, equation (32).",
    "formula-diagonal": "$r(k,k)=\\dfrac2\\pi\\sum_{j=1}^{k}\\dfrac1{2j-1}$ for integers $k\\geq1$, as follows from the diagonal recurrence in CITE{Cserti}, equation (32).",
    "formula-small": "$r(1,0)=\\tfrac12$, $r(1,1)=\\tfrac2\\pi$, $r(2,0)=2-\\tfrac4\\pi$, and $r(2,1)=\\tfrac4\\pi-\\tfrac12$ CITE{Cserti}, equations (26), (29), and (32)."
  },
  "Similar tables": [
    {
      "table": "HREF{Kirchhoff_indices_of_connected_graphs}[Kirchhoff indices of connected graphs]",
      "relation": "sums effective resistances over vertex pairs in finite connected graphs"
    },
    {
      "table": "HREF{Watson_integrals_of_the_cubic_lattices}[Watson integrals of the cubic lattices]",
      "relation": "holds convergent origin lattice Green-function integrals in three dimensions"
    }
  ],
  "Programs": {
    "program-mpmath": {
      "language": "Python",
      "code": "import mpmath as mp\nmp.mp.dps = 120\n\ndef resistance(m, n):\n    # Numerical quadrature; the attached generator provides certified bounds.\n    def integrand(y):\n        if not y:\n            return mp.mpf(m)\n        s = 2*mp.asinh(mp.sin(y/2))\n        decay = mp.exp(-m*s)\n        return (-mp.expm1(-m*s) + 2*decay*mp.sin(n*y/2)**2)/mp.sinh(s)\n    return mp.quad(integrand, [0, mp.pi/4, mp.pi/2, mp.pi])/mp.pi\n\nprint(mp.nstr(resistance(2, 1), 100))"
    }
  }
}