{
  "model": "Nearest-neighbour simple symmetric random walk on the integer lattice Z^d",
  "generatedAt": "2026-09-11T12:00:00+02:00",
  "maximumPairs": 100000,
  "maximumSteps": 200000,
  "method": {
    "oneDimension": "p_1(2n) = C(2n,n) / 4^n, updated by p_n = p_(n-1)(2n-1)/(2n)",
    "twoDimensions": "p_2(2n) = p_1(2n)^2",
    "threeDimensions": "p_3(2n) = p_1(2n) b_n, where b_n = a_n/9^n and n^2 a_n = (10n^2-10n+3)a_(n-1)-9(n-1)^2a_(n-2)",
    "tail": "First-order local-central-limit tail: sum_(n>N) p_3(2n) approximately 2c/sqrt(N), c = 2(3/(4pi))^(3/2)"
  },
  "threeDimensions": {
    "partialGreen": 1.5149106059175197,
    "localLimitCoefficient": 0.233290514929399,
    "firstOrderTail": 0.0014754587673808323,
    "extrapolatedGreen": 1.5163860646849006,
    "extrapolatedReturnProbability": 0.34053733195721747,
    "publishedComparison": 0.3405373296,
    "absoluteDifference": 2.357217476767204e-9
  },
  "checkpoints": [
    {
      "n": 1,
      "steps": 2,
      "returnProbability1d": 0.5,
      "returnProbability2d": 0.25,
      "returnProbability3d": 0.16666666666666666,
      "partialGreen1d": 1.5,
      "partialGreen2d": 1.25,
      "partialGreen3d": 1.1666666666666667
    },
    {
      "n": 10,
      "steps": 20,
      "returnProbability1d": 0.17619705200195312,
      "returnProbability2d": 0.031045401134178974,
      "returnProbability3d": 0.007108354149086694,
      "partialGreen1d": 3.7001380920410156,
      "partialGreen2d": 1.822389342720271,
      "partialGreen3d": 1.374121551459555
    },
    {
      "n": 100,
      "steps": 200,
      "returnProbability1d": 0.056348479009256436,
      "returnProbability2d": 0.003175151086656613,
      "returnProbability3d": 0.00023241805105917422,
      "partialGreen1d": 11.326044280860545,
      "partialGreen2d": 2.5345272637222265,
      "partialGreen3d": 1.4699021030758814
    },
    {
      "n": 1000,
      "steps": 2000,
      "returnProbability1d": 0.017839011145854306,
      "returnProbability2d": 0.00031823031866191415,
      "returnProbability3d": 0.000007374528101166391,
      "partialGreen1d": 35.69586130285449,
      "partialGreen2d": 3.2653213109910864,
      "partialGreen3d": 1.5016370018443337
    },
    {
      "n": 10000,
      "steps": 20000,
      "returnProbability1d": 0.005641825312220405,
      "returnProbability2d": 0.00003183019285361087,
      "returnProbability3d": 2.3328176677201593e-7,
      "partialGreen1d": 112.8421480697207,
      "partialGreen2d": 3.998042121255224,
      "partialGreen3d": 1.5117204238130428
    },
    {
      "n": 100000,
      "steps": 200000,
      "returnProbability1d": 0.0017841218859990715,
      "returnProbability2d": 0.000003183090904100884,
      "returnProbability3d": 7.3772661721229464e-9,
      "partialGreen1d": 356.82616132169306,
      "partialGreen2d": 4.730956234923081,
      "partialGreen3d": 1.5149106059175197
    }
  ]
}