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CellChangeTimes->{{3.520176659327881*^9, 3.520176681907336*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Total", "[", "MatrixElementBinCounts", "]"}]], "Input"], Cell[BoxData["40000"], "Output", CellChangeTimes->{3.520176691398387*^9}] }, Open ]], Cell[TextData[{ "On this evidence, the probability that a matrix element will be found in \ the ", Cell[BoxData[ FormBox[ SuperscriptBox["k", "th"], TraditionalForm]], FormatType->"TraditionalForm"], " bin is" }], "Text", CellChangeTimes->{{3.520176745514658*^9, 3.5201767842538233`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"Prob", "[", "k_", "]"}], ":=", FractionBox[ RowBox[{ "MatrixElementBinCounts", "\[LeftDoubleBracket]", "k", "\[RightDoubleBracket]"}], "40000"]}]], "Input", CellChangeTimes->{{3.5201298855877943`*^9, 3.520129894073339*^9}, { 3.5201299253908*^9, 3.520129940807357*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Sum", "[", RowBox[{ RowBox[{"Prob", "[", "k", "]"}], ",", RowBox[{"{", RowBox[{"k", ",", "1", ",", "120"}], 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CellChangeTimes->{3.520172351278852*^9}] }, Open ]], Cell[TextData[{ "that if X and Y are independent random variable drawn from normal \ distributions with standard deviations ", Cell[BoxData[ SubscriptBox["\[Sigma]", "1"]], CellChangeTimes->{ 3.5201097402638407`*^9, {3.5201784111206713`*^9, 3.5201784128251877`*^9}}], " and ", Cell[BoxData[ SubscriptBox["\[Sigma]", "2"]], CellChangeTimes->{ 3.5201097402638407`*^9, {3.5201784111206713`*^9, 3.5201784128251877`*^9}, 3.520178449365285*^9}], " then we expect sums X + Y to be normally distributed with ", Cell[BoxData[ RowBox[{"\[Sigma]", "=", SqrtBox[ RowBox[{ SuperscriptBox[ SubscriptBox["\[Sigma]", "1"], "2"], "+", SuperscriptBox[ SubscriptBox["\[Sigma]", "2"], "2"]}]]}]], CellChangeTimes->{{3.520178547031982*^9, 3.520178574416395*^9}}], ", and therefore expect the averaged sums ", Cell[BoxData[ FractionBox[ RowBox[{"X", "+", "Y"}], "2"]], CellChangeTimes->{{3.520178660760985*^9, 3.5201786695807943`*^9}}], "to be normally distributed 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\[Sigma] \[Rule] 0.706164. \ We conclude that ", StyleBox["the procedure described above does in fact produce symmetric \ matrices with normally (and identically) distributed matrix elements", "Subsection", FontSize->12], "\[Ellipsis]the statistical independence of which is compromised only by the \ symmetry constraint." }], "Text", CellChangeTimes->{{3.520178965483639*^9, 3.520178985122649*^9}, 3.5201790266643667`*^9, {3.520179062246077*^9, 3.520179211830309*^9}}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.520179250150798*^9}] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Eigenvalues of symmetric matrices with normally distributed random elements\ \>", "Subsection", CellChangeTimes->{{3.5201792974800262`*^9, 3.520179332629533*^9}}], Cell[CellGroupData[{ Cell["Large population of small matrices", "Subsubsection", CellChangeTimes->{{3.520181178515827*^9, 3.520181188624138*^9}, { 3.5201816334789143`*^9, 3.520181635998096*^9}}], Cell["\<\ We look first to the 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In the present instance (1000 \[Times] 1000 matrix with normally \ distributed matrix elements, standard deviation ", Cell[BoxData[ RowBox[{"\[Sigma]", "=", RowBox[{"1", "/", SqrtBox["2"]}]}]], CellChangeTimes->{{3.520253757739856*^9, 3.520253773713719*^9}}], " ", "= 0.707107", ") we found" }], "Text", CellChangeTimes->{{3.520253409597776*^9, 3.520253432267931*^9}, { 3.520253493160471*^9, 3.520253520021215*^9}, {3.520253585518214*^9, 3.520253609668272*^9}, {3.520253645880101*^9, 3.520253654705123*^9}, { 3.5202536880128927`*^9, 3.520253736001906*^9}, {3.520253795897838*^9, 3.520253796489963*^9}, {3.5202538670180893`*^9, 3.5202538986481113`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"Wigner", " ", "radius"}], " ", "=", " ", RowBox[{ RowBox[{"largest", " ", "eigenvalue"}], "=", "44.7472"}]}]], "DisplayFormula", CellChangeTimes->{{3.52025391235993*^9, 3.520253968772561*^9}}], Cell["in which connection it is interesting to notice that", "Text", CellChangeTimes->{{3.520254027990981*^9, 3.520254050645944*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"N", "[", SqrtBox[ RowBox[{"2", "*", "1000"}]], "]"}]], "Input", CellChangeTimes->{{3.520254064520756*^9, 3.5202540909019136`*^9}}], Cell[BoxData["44.721359549995796`"], "Output", CellChangeTimes->{3.5202540989673443`*^9}] }, Open ]], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.520254138825739*^9}] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Conclusions drawn from numerical data now in hand", "Subsection", CellChangeTimes->{{3.520254148339007*^9, 3.520254163398036*^9}}], Cell[TextData[{ "Let the elements of the N-dimensional square matrix \[DoubleStruckCapitalM] \ be normally distributed with ", Cell[BoxData[ FormBox[ SubscriptBox["\[Sigma]", "M"], TraditionalForm]], FormatType->"TraditionalForm"], " = 1. Then\n" }], "Text", CellChangeTimes->{{3.520254315644209*^9, 3.520254429174449*^9}, { 3.520254698448081*^9, 3.5202547049638643`*^9}, 3.520254956149659*^9}], Cell[TextData[{ "\[Bullet] The elements of the N-dimensional symmetric matrix ", Cell[BoxData[ RowBox[{"\[DoubleStruckCapitalS]", "=", FractionBox[ RowBox[{"\[DoubleStruckCapitalM]", "+", SuperscriptBox["\[DoubleStruckCapitalM]", "transpose"]}], "2"]}]], CellChangeTimes->{{3.520254598109495*^9, 3.520254599355907*^9}}], " (\[DoubleStruckCapitalS] is simply the ", StyleBox["symmetric part", FontVariations->{"Underline"->True}], " of \[DoubleStruckCapitalM]) are normally distributed with ", Cell[BoxData[ RowBox[{ SubscriptBox["\[Sigma]", "S"], "=", RowBox[{"1", "/", SqrtBox["2"]}]}]], CellChangeTimes->{{3.520253757739856*^9, 3.520253773713719*^9}}], ". More generally, we have" }], "Text", CellChangeTimes->{{3.520254508683535*^9, 3.5202545438978558`*^9}, { 3.520254616772011*^9, 3.520254656326727*^9}, {3.520254714760846*^9, 3.520254755614476*^9}, {3.520262554660102*^9, 3.5202625788944283`*^9}}], Cell[BoxData[ RowBox[{ SubscriptBox["\[Sigma]", "S"], "=", RowBox[{ FractionBox[ SqrtBox[ RowBox[{ SuperscriptBox[ SubscriptBox["\[Sigma]", "M"], "2"], "+", SuperscriptBox[ SubscriptBox["\[Sigma]", "M"], "2"]}]], "2"], "=", FractionBox[ SubscriptBox["\[Sigma]", "M"], SqrtBox["2"]]}]}]], "DisplayFormula", CellChangeTimes->{{3.5202547638231*^9, 3.520254894557639*^9}}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.520254942843471*^9}], Cell["\<\ \[Bullet] The eigenvalues of \[DoubleStruckCapitalS] can be expected to \ conform well to Wigner's Semicircular Law if N > about 1000, but for N \ \[LessLess] 1000 they conform to N-dependent \"Sombrero Laws\" for which we \ possess as yet no analytic description.\ \>", "Text", CellChangeTimes->{ 3.5202549906732693`*^9, {3.520255035890195*^9, 3.5202550932682*^9}, { 3.520255192529298*^9, 3.5202552181494226`*^9}, {3.520255260907135*^9, 3.520255268121294*^9}, {3.520255334383894*^9, 3.520255370051265*^9}}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.5202554364308662`*^9}], Cell[TextData[{ "\[Bullet] The spectrum of \[DoubleStruckCapitalS] is\[LongDash]for N \ sufficiently large\[LongDash]bounded (exactly? in good approximation?) by ", Cell[BoxData[ RowBox[{"\[PlusMinus]", SqrtBox[ RowBox[{"2", "N"}]]}]], CellChangeTimes->{{3.520256464879815*^9, 3.5202564669080048`*^9}}], "." }], "Text", CellChangeTimes->{{3.5202554702725143`*^9, 3.520255486206772*^9}, { 3.520256341642646*^9, 3.520256427893488*^9}, {3.520256508771781*^9, 3.520256512086947*^9}}], Cell[BoxData["\[IndentingNewLine]"], "Input", CellChangeTimes->{3.520256535218474*^9}] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Effect of adjusting the value of \[Sigma]\ \>", "Subsection", CellChangeTimes->{{3.5202565456839743`*^9, 3.5202565564041643`*^9}}], Cell["\<\ I look specifically the consequences of the adjustment \[Sigma] = 1 \ \[RightArrow] 2.\ \>", "Text", CellChangeTimes->{{3.520256578523883*^9, 3.520256601420573*^9}, { 3.5202566416293373`*^9, 3.520256644946459*^9}}], Cell["\<\ REMARK: I wish I had assigned a new name to this list. 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